A125276 -id:A125276 - cp's OEIS frontend

A125275 Eigensequence of triangle A039599: a(n) = Sum_{k=0..n-1} A039599(n-1,k)*a(k) for n > 0 with a(0) = 1.

1, 1, 2, 7, 31, 162, 968, 6481, 47893, 386098, 3364562, 31460324, 313743665, 3320211313, 37124987124, 436985496790, 5397178181290, 69748452377058, 940762812167126, 13213888481979449, 192891251215160017

Offset: 0

Paul D. Hanna, Nov 26 2006

nonn

Starting with offset 1, these are the row sums of triangle A147294. - Gary W. Adamson, Nov 05 2008

			a(3) = 2*(1) + 3*(1) + 1*(2) = 7;
a(4) = 5*(1) + 9*(1) + 5*(2) + 1*(7) = 31;
a(5) = 14*(1) + 28*(1) + 20*(2) + 7*(7) + 1*(31) = 162.
Triangle A039599(n,k) = C(2*n+1, n-k)*(2*k+1)/(2*n+1) (with rows n >= 0 and columns k = 0..n) begins:
   1;
   1,  1;
   2,  3,  1;
   5,  9,  5,  1;
  14, 28, 20,  7, 1;
  42, 90, 75, 35, 9, 1;
  ...
where the g.f. of column k is G000108(x)^(2*k+1)
and G000108(x) = (1 - sqrt(1 - 4*x))/(2*x) is the Catalan g.f. function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..500
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Cf. A000108, A039599, A125276 (variant), A147294.

A125275=ConstantArray[0,20]; A125275[[1]]=1; Do[A125275[[n]]=Binomial[2*n-1,n-1]/(2*n-1)+Sum[A125275[[k]]*Binomial[2*n-1,n-k-1]*(2*k+1)/(2*n-1),{k,1,n-1}];,{n,2,20}]; Flatten[{1,A125275}] (* Vaclav Kotesovec, Dec 09 2013 *)

a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(2*n-1, n-k-1)*(2*k+1)/(2*n-1)))

a(n) = Sum_{k=0..n-1} a(k) * C(2*n-1, n-k-1) * (2*k + 1)/(2*n - 1) for n > 0 with a(0) = 1.

A132289 Triangle, read by rows of 2n+1 terms, where T(n,k) = T(n,k-1) + T(n-1,k-1) for 2n>=k>0, T(n,2n-1) = T(n,2n-2) + T(n-1,n-1) and T(n,2n) = T(n,2n-1) + T(n-1,n-1) for n>0, with T(n,0) = T(n-1,n-1) for n>0 and T(0,0) = 1.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 6, 9, 12, 12, 15, 19, 25, 34, 46, 58, 58, 70, 85, 104, 129, 163, 209, 267, 325, 325, 383, 453, 538, 642, 771, 934, 1143, 1410, 1735, 2060, 2060, 2385, 2768, 3221, 3759, 4401, 5172, 6106, 7249, 8659, 10394, 12454, 14514, 14514, 16574, 18959

Offset: 0

Paul D. Hanna, Aug 18 2007

Column 0 and the rightmost border (shifted right) equal A125276.

			Triangle begins:
  1;
  1, 2, 3;
  3, 4, 6, 9, 12;
  12, 15, 19, 25, 34, 46, 58;
  58, 70, 85, 104, 129, 163, 209, 267, 325;
  325, 383, 453, 538, 642, 771, 934, 1143, 1410, 1735, 2060;
  2060, 2385, 2768, 3221, 3759, 4401, 5172, 6106, 7249, 8659, 10394, 12454, 14514; ...

Cf. A132290 (main diagonal), A125276 (column 0).

PARI
```
T(n,k)=my(A=[1]);if(2*n
				
```

A132290 Main diagonal of triangle A132289: a(n) = A132289(n,n) for n>=0.

Original entry on oeis.org

1, 2, 6, 25, 129, 771, 5172, 38280, 308987, 2695586, 25231269, 251869681, 2667876382, 29856632341, 351717286357, 4347357161694, 56222019376378, 758836553993985, 10665452531786380, 155785594704483911

Offset: 0

Paul D. Hanna, Aug 18 2007

nonn

Equals the binomial transform of A125276.

Cf. A132289, A125276.

{a(n)=local(A=[1]);if(n==0,1,for(i=1,n, A=Vec(Ser(concat(concat(A[ #A],A),A[ #A]))/(1-x)));A[n+1])}

G.f. A(x) satisfies: A( x*(1-x)/(1+x-x^2) )/(1+x-x^2) = 1 + x*A(x).

cp's OEIS Frontend

A125275 Eigensequence of triangle A039599: a(n) = Sum_{k=0..n-1} A039599(n-1,k)*a(k) for n > 0 with a(0) = 1.

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A132289 Triangle, read by rows of 2n+1 terms, where T(n,k) = T(n,k-1) + T(n-1,k-1) for 2n>=k>0, T(n,2n-1) = T(n,2n-2) + T(n-1,n-1) and T(n,2n) = T(n,2n-1) + T(n-1,n-1) for n>0, with T(n,0) = T(n-1,n-1) for n>0 and T(0,0) = 1.

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A132290 Main diagonal of triangle A132289: a(n) = A132289(n,n) for n>=0.

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