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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, 140681, 19303, 1821, 137, 10, 1 (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
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;
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
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Oct 31 2005
STATUS
approved

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Last modified March 28 13:35 EDT 2024. Contains 371254 sequences. (Running on oeis4.)