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 A112705 Triangle built from partial sums of Catalan numbers A000108 multiplied by powers. 12
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 W. 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=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; 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; ... 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

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Last modified April 5 23:13 EDT 2020. Contains 333260 sequences. (Running on oeis4.)