A172171 - OEIS

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A172171

(1, 9) Pascal Triangle read by horizontal rows. Same as A093644, but mirrored and without the additional row/column (1, 9, 9, 9, 9, ...).

1, 1, 10, 1, 11, 19, 1, 12, 30, 28, 1, 13, 42, 58, 37, 1, 14, 55, 100, 95, 46, 1, 15, 69, 155, 195, 141, 55, 1, 16, 84, 224, 350, 336, 196, 64, 1, 17, 100, 308, 574, 686, 532, 260, 73, 1, 18, 117, 408, 882, 1260, 1218, 792, 333, 82

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

OFFSET

1,3

COMMENTS

Binomial transform of A017173.

LINKS

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

FORMULA

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(n,1) = 1, T(2,2) = 10, T(n,k) = 0 if k < 1 or if k > n.

Sum_{k=0..n} T(n, k) = A139634(n).

T(2*n-1, n) = A050489(n).

EXAMPLE

Triangle begins:

1, 10;

1, 11, 19;

1, 12, 30, 28;

1, 13, 42, 58, 37;

1, 14, 55, 100, 95, 46;

1, 15, 69, 155, 195, 141, 55;

1, 16, 84, 224, 350, 336, 196, 64;

1, 17, 100, 308, 574, 686, 532, 260, 73;

1, 18, 117, 408, 882, 1260, 1218, 792, 333, 82;

1, 19, 135, 525, 1290, 2142, 2478, 2010, 1125, 415, 91;

1, 20, 154, 660, 1815, 3432, 4620, 4488, 3135, 1540, 506, 100;

MATHEMATICA

T[n_, k_]:= T[n, k]= If[k<1 || k>n, 0, If[k==1, 1, If[n==2 && k==2, 10, T[n-1, k] + 2*T[n-1, k-1] - T[n-2, k-1] - T[n-2, k-2]]]];

Table[T[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Apr 24 2022 *)

PROG

(SageMath)

@CachedFunction

def T(n, k):

if (k<1 or k>n): return 0

elif (k==1): return 1

elif (n==2 and k==2): return 10

else: return T(n-1, k) + 2*T(n-1, k-1) - T(n-2, k-1) - T(n-2, k-2)

flatten([[T(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 24 2022

CROSSREFS

Cf. A007318, A017173, A050489 (central terms), A093644, A139634 (row sums).

Sequence in context: A297418 A297352 A370401 * A327723 A164899 A164844

Adjacent sequences: A172168 A172169 A172170 * A172172 A172173 A172174

KEYWORD

nonn,tabl

AUTHOR

Mark Dols, Jan 28 2010

EXTENSIONS

More terms from Philippe Deléham, Dec 25 2013

STATUS

approved