A112705 - OEIS

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A112705

Triangle built from partial sums of Catalan numbers A000108 multiplied by powers.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 11, 4, 1, 1, 23, 51, 22, 5, 1, 1, 65, 275, 157, 37, 6, 1, 1, 197, 1619, 1291, 357, 56, 7, 1, 1, 626, 10067, 11497, 3941, 681, 79, 8, 1, 1, 2056, 64979, 107725, 46949, 9431, 1159, 106, 9, 1, 1, 6918, 431059, 1045948, 587621, 140681, 19303, 1821, 137, 10, 1

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OFFSET

0,5

COMMENTS

The column sequences (without leading zeros) begin with A000012 (powers of 1), A112705 (partial sums Catalan), A112696-A112704, for m=0..10.

LINKS

Table of n, a(n) for n=0..65.

Wolfdieter Lang, First 10 rows.

FORMULA

a(n, m) = sum(C(k)*m^k, k=0..n-m), n>m>0, with C(n):=A000108(n); a(n, n)=1; a(n, 0)=1; a(n, m)=0 if n<m.

G.f. for column m>=0 (without leading zeros): c(m*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.

EXAMPLE

Triangle starts:

1, 1;

1, 2, 1;

1, 4, 3, 1;

1, 9, 11, 4, 1;

1, 23, 51, 22, 5, 1;

1, 65, 275, 157, 37, 6, 1;

...

MATHEMATICA

col[m_] := col[m] = CatalanNumber[#]*m^#& /@ Range[0, 20] // Accumulate;

T[n_, m_] := If[m == 0, 1, col[m][[n - m + 1]]];

Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Aug 29 2022 *)

PROG

(PARI) t(n, m) = if (m==0, 1, if (n==m, 1, sum(kk=0, n-m, m^kk*binomial(2*kk, kk)/(kk+1))));

tabl(nn) = {for (n=0, nn, for (m=0, n, print1(t(n, m), ", "); ); print(); ); } \\ Michel Marcus, Nov 25 2015

CROSSREFS

Row sums give A112706.

Sequence in context: A091150 A091351 A058730 * A070895 A127054 A125790

Adjacent sequences: A112702 A112703 A112704 * A112706 A112707 A112708

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Oct 31 2005

STATUS

approved