A304187 -id:A304187

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A101479

Triangular matrix T, read by rows, where row n equals row (n-1) of T^(n-1) after appending '1' for the main diagonal.

+10
25

1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 19, 9, 3, 1, 1, 191, 70, 18, 4, 1, 1, 2646, 795, 170, 30, 5, 1, 1, 46737, 11961, 2220, 335, 45, 6, 1, 1, 1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1, 25330125, 5051866, 758814, 92652, 9730, 924, 84, 8, 1, 1, 735180292, 132523155, 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

Remarkably, T equals the product of these triangular matrices: T = A107867*A107862^-1 = A107870*A107867^-1 = A107873*A107870^-1; reversing the order of these products yields triangle A107876.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened (first 31 rows from Paul D. Hanna)

EXAMPLE

Triangle begins:

1, 1;

1, 1, 1;

3, 2, 1, 1;

19, 9, 3, 1, 1;

191, 70, 18, 4, 1, 1;

2646, 795, 170, 30, 5, 1, 1;

46737, 11961, 2220, 335, 45, 6, 1, 1;

1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1;

25330125, 5051866, 758814, 92652, 9730, 924, 84, 8, 1, 1;

735180292, 132523155, 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1; ...

Row 4 starts with row 3 of T^3 which begins:

3, 1;

6, 3, 1;

19, 9, 3, 1; ...

row 5 starts with row 4 of T^4 which begins:

4, 1;

10, 4, 1;

34, 14, 4, 1;

191, 70, 18, 4, 1; ...

An ALTERNATE GENERATING METHOD is illustrated as follows.

For row 4:

Start with a '1' and append 2 zeros,

take partial sums and append 1 zero,

take partial sums thrice more, resulting in:

1, 0, 0;

1, 1, 1, 0;

1, 2, 3, 3;

1, 3, 6, 9;

1, 4,10,19.

Final nonzero terms form row 4: [19,9,3,1,1].

For row 5:

Start with a '1' and append 3 zeros,

take partial sums and append 2 zeros,

take partial sums and append 1 zero,

take partial sums thrice more, resulting in:

1, 0, 0, 0;

1, 1, 1, 1, 0, 0;

1, 2, 3, 4, 4, 4, 0;

1, 3, 6,10,14, 18, 18;

1, 4,10,20,34, 52, 70;

1, 5,15,35,69,121,191;

where the final nonzero terms form row 5: [191,70,18,4,1,1].

Likewise, for row 6:

1, 0, 0, 0, 0;

1, 1, 1, 1, 1, 0, 0, 0;

1, 2, 3, 4, 5, 5, 5, 5, 0, 0;

1, 3, 6,10, 15, 20, 25, 30, 30, 30, 0;

1, 4,10,20, 35, 55, 80,110, 140, 170, 170;

1, 5,15,35, 70,125,205,315, 455, 625, 795;

1, 6,21,56,126,251,456,771,1226,1851,2646;

where the final nonzero terms form row 6: [2646,795,170,30,5,1,1].

Continuing in this way generates all rows of this triangle.

MAPLE

b:= proc(n) option remember;

Matrix(n, (i, j)-> T(i-1, j-1))^(n-1)

end:

T:= proc(n, k) option remember;

`if`(n=k, 1, `if`(k>n, 0, b(n)[n, k+1]))

end:

seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Apr 13 2020

MATHEMATICA

b[n_] := b[n] = MatrixPower[Table[T[i-1, j-1], {i, n}, {j, n}], n-1];

T[n_, k_] := T[n, k] = If[n == k, 1, If[k > n, 0, b[n][[n, k+1]]]];

Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 25 2020, after Alois P. Heinz *)

PROG

(PARI) {T(n, k) = my(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j] = (A^(i-2))[i-1, j]); )); A=B); return(A[n+1, k+1])}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

(PARI) {T(n, k) = my(A=vector(n+1), p); A[1]=1; for(j=1, n-k-1, p=(n-1)*(n-2)/2-(n-j-1)*(n-j-2)/2; A = Vec((Polrev(A)+x*O(x^p))/(1-x))); A = Vec((Polrev(A) +x*O(x^p)) / (1-x) ); A[p+1]}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Columns are A101481, A101482, A101483, row sums form A101484.

Cf. A107876 (dual triangle).

Cf. A304184, A304185, A304186, A304187.

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jan 21 2005, Jul 26 2006, May 27 2007

STATUS

approved

A304184

G.f. A(x) satisfies: 0 = [x^n] (1+x)^(n*(n-1)/2) / A(x) for n>0.

+10
5

1, 0, 0, 1, 9, 117, 1851, 34923, 765933, 19155084, 538051164, 16771165230, 574424285076, 21443516818065, 866521903003641, 37683366660458208, 1754777541925339779, 87115221430910051901, 4592968693335470802627, 256294382115032521083411, 15090698035153332532531074

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..20.

FORMULA

[x^n] (1+x)^(n*(n+1)/2) / A(x) = A101481(n+1) = A101479(n+1,0) for n>=0.

[x^n] (1+x)^((n+1)*(n+2)/2) / A(x) = Sum_{k=0..n} A101479(n+2,k+1) * A101479(k+1,0) for n>=0.

EXAMPLE

G.f.: A(x) = 1 + x^3 + 9*x^4 + 117*x^5 + 1851*x^6 + 34923*x^7 + 765933*x^8 + 19155084*x^9 + 538051164*x^10 + 16771165230*x^11 + 574424285076*x^12 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k in (1+x)^(n*(n-1)/2) / A(x) begins:

n=0: [1, 0, 0, -1, -9, -117, -1850, -34905, -765618, ...];

n=1: [1, 0, 0, -1, -9, -117, -1850, -34905, -765618, ...];

n=2: [1, 1, 0, -1, -10, -126, -1967, -36755, -800523, ...];

n=3: [1, 3, 3, 0, -12, -147, -2229, -40815, -876000, ...];

n=4: [1, 6, 15, 19, 0, -180, -2706, -47955, -1005279, ...];

n=5: [1, 10, 45, 119, 191, 0, -3335, -59840, -1214055, ...];

n=6: [1, 15, 105, 454, 1341, 2646, 0, -73965, -1545531, ...];

n=7: [1, 21, 210, 1329, 5955, 19833, 46737, 0, -1913457, ...];

n=8: [1, 28, 378, 3275, 20438, 97533, 364936, 1003150, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] (1+x)^(n*(n-1)/2) / A(x) for n>0.

RELATED SEQUENCES.

The secondary diagonal in the above table that begins

[1, 1, 3, 19, 191, 2646, 46737, 1003150, 25330125, ...]

yields A101481, column 0 of triangle A101479.

Related triangular matrix T = A101479 begins:

1, 1;

1, 1, 1;

3, 2, 1, 1;

19, 9, 3, 1, 1;

191, 70, 18, 4, 1, 1;

2646, 795, 170, 30, 5, 1, 1;

46737, 11961, 2220, 335, 45, 6, 1, 1;

1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...

in which row n equals row (n-1) of T^(n-1) followed by '1' for n>0.

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^((m-1)*(m-2)/2)/Ser(A) )[m] ); A[n+1]}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A101481, A304185, A304186, A304187, A304189, A101479.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 08 2018

STATUS

approved

A304185

G.f. A(x) satisfies: 0 = [x^n] (1+x)^(n*(n+1)/2) / A(x) for n>0.

+10
4

1, 1, 1, 6, 51, 609, 9284, 171779, 3729929, 92828134, 2602268335, 81082146565, 2778537934965, 103826098384590, 4200631499008965, 182912327481135887, 8528448938007615767, 423908532350099458532, 22375155669520993773517, 1249871928128324730985042, 73662719257076005954584046

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..20.

FORMULA

[x^n] (1+x)^((n+1)*(n+2)/2) / A(x) = A101482(n+1) = A101479(n+2,1) for n>=0.

[x^n] (1+x)^((n+2)*(n+3)/2) / A(x) = Sum_{k=0..n} A101479(n+3,k+2) * A101479(k+2,1) for n>=0.

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 51*x^4 + 609*x^5 + 9284*x^6 + 171779*x^7 + 3729929*x^8 + 92828134*x^9 + 2602268335*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k in (1+x)^(n*(n+1)/2) / A(x) begins:

n=0: [1, -1, 0, -5, -40, -513, -8092, -153395, -3388500, ...];

n=1: [1, 0, -1, -5, -45, -553, -8605, -161487, -3541895, ...];

n=2: [1, 2, 0, -7, -56, -648, -9756, -179250, -3873474, ...];

n=3: [1, 5, 9, 0, -75, -837, -11875, -210518, -4441140, ...];

n=4: [1, 9, 35, 70, 0, -1096, -15664, -263340, -5357885, ...];

n=5: [1, 14, 90, 345, 795, 0, -20260, -352235, -6842115, ...];

n=6: [1, 20, 189, 1115, 4510, 11961, 0, -452166, -9245340, ...];

n=7: [1, 27, 350, 2893, 17019, 74282, 224504, 0, -11809259, ...];

n=8: [1, 35, 594, 6505, 51545, 312984, 1483340, 5051866, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] (1+x)^(n*(n+1)/2) / A(x) for n>0.

RELATED SEQUENCES.

The secondary diagonal in the above table that begins

[1, 2, 9, 70, 795, 11961, 224504, 5051866, 132523155, ...]

yields A101482, column 1 of triangle A101479.

Related triangular matrix T = A101479 begins:

1, 1;

1, 1, 1;

3, 2, 1, 1;

19, 9, 3, 1, 1;

191, 70, 18, 4, 1, 1;

2646, 795, 170, 30, 5, 1, 1;

46737, 11961, 2220, 335, 45, 6, 1, 1;

1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...

in which row n equals row (n-1) of T^(n-1) followed by '1' for n>0.

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^(m*(m-1)/2)/Ser(A) )[m] ); A[n+1]}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A101481, A304184, A304186, A304187, A101479.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 08 2018

STATUS

approved

A304186

G.f. A(x) satisfies: 0 = [x^n] (1+x)^((n+1)*(n+2)/2) / A(x) for n>0.

+10
4

1, 3, 6, 24, 189, 2199, 33495, 625743, 13778469, 348258723, 9916111584, 313642702743, 10901017499472, 412730651160567, 16902257604014685, 744247858158013245, 35058132248539742325, 1758994171367292095805, 93646661943861501833100, 5272500660870261306912750

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..19.

FORMULA

[x^n] (1+x)^((n+2)*(n+3)/2) / A(x) = A101483(n+1) = A101479(n+3,2) for n>=0.

[x^n] (1+x)^((n+3)*(n+4)/2) / A(x) = Sum_{k=0..n} A101479(n+4,k+3) * A101479(k+3,2) for n>=0.

EXAMPLE

G.f.: A(x) = 1 + 3*x + 6*x^2 + 24*x^3 + 189*x^4 + 2199*x^5 + 33495*x^6 + 625743*x^7 + 13778469*x^8 + 348258723*x^9 + 9916111584*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k in (1+x)^((n+1)*(n+2)/2) / A(x) begins:

n=0: [1, -2, 0, -12, -105, -1434, -23877, -473730, -10881882, ...];

n=1: [1, 0, -3, -14, -129, -1656, -26850, -522918, -11853219, ...];

n=2: [1, 3, 0, -22, -180, -2088, -32219, -608565, -13504179, ...];

n=3: [1, 7, 18, 0, -255, -2937, -41739, -750711, -16140285, ...];

n=4: [1, 12, 63, 170, 0, -3996, -58877, -991308, -20341875, ...];

n=5: [1, 18, 150, 748, 2220, 0, -78435, -1401570, -27251715, ...];

n=6: [1, 25, 297, 2211, 11271, 37149, 0, -1843458, -38615364, ...];

n=7: [1, 33, 525, 5343, 38745, 207663, 758814, 0, -50361381, ...];

n=8: [1, 42, 858, 11340, 108630, 797100, 4541805, 18301950, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] (1+x)^((n+1)*(n+2)/2) / A(x) for n>0.

RELATED SEQUENCES.

The secondary diagonal in the above table that begins

[1, 3, 18, 170, 2220, 37149, 758814, 18301950, 508907970, ...]

yields A101483, column 2 of triangle A101479.

Related triangular matrix T = A101479 begins:

1, 1;

1, 1, 1;

3, 2, 1, 1;

19, 9, 3, 1, 1;

191, 70, 18, 4, 1, 1;

2646, 795, 170, 30, 5, 1, 1;

46737, 11961, 2220, 335, 45, 6, 1, 1;

1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...

in which row n equals row (n-1) of T^(n-1) followed by '1' for n>0.

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^(m*(m+1)/2)/Ser(A) )[m] ); A[n+1]}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A101481, A304184, A304185, A304187, A101479.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 08 2018

STATUS

approved

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