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

Offset: 0

Author

Wolfdieter Lang, Oct 31 2005

Keywords

• nonn
• easy
• tabl

Comments

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

Examples

			Triangle starts:
  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;
  ...
		

Links

• Wolfdieter Lang, First 10 rows.

Crossrefs

Row sums give A112706.

Programs

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 *)

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

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

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.

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