A125275 -id:A125275 - cp's OEIS frontend

A147294 Eigentriangle, row sums = A125275.

1, 1, 1, 2, 3, 2, 5, 9, 10, 7, 14, 28, 40, 49, 31, 42, 90, 150, 245, 279, 162, 132, 297, 550, 1078, 1674, 1782, 968, 429, 1001, 2002, 4459, 8463, 12474, 12584, 6481, 1430, 3432, 7280, 17836, 39060, 71280, 100672, 97215, 47893

Offset: 1

Gary W. Adamson, Nov 05 2008

Row sums = A125275 starting with offset 1: (1, 2, 7, 31, 162, 968,...).
Left border = A000108, right border = A125275.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
2, 3, 2;
5, 9, 10, 7;
14, 28, 40, 49, 31;
42, 90, 150, 245, 279, 162;
132, 297, 550, 1078, 1674, 1782, 968;
429, 1001, 2002, 4459, 8463, 12474, 12584, 6481;
1430, 3432, 7280, 17836, 39060, 71280, 100672, 97215, 47893;
...
Row 4 = (5, 9, 10, 7) = termwise products of (5, 9, 5, 1) and (1, 1, 2, 7)

A125275, Cf. A000108, A039599

A147294 = Triangle A039599 * (A125275 * 0^(n-k)); where (A125275 * 0^(n-k)) = an infinite lower triangular matrix with A125275: (1, 1, 2, 7, 31, 162,...) as the main diagonal and the rest zeros.

A144250 Eigentriangle, row sums = A125275, shifted.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 23, 1, 15, 70, 168, 207, 106, 1, 21, 140, 504, 1035, 1166, 567, 1, 28, 252, 1260, 3795, 6996, 7371, 3434

Offset: 0

Gary W. Adamson, Sep 16 2008

Row sums = A125273 shifted. A125273 = the eigensequence of triangle A085478.

Right border = A125273: (1, 1, 2, 6, 23, 106, 567, 3434,...). Sum of n-th row terms = rightmost term in next row.

			First few rows of the triangle =
1;
1, 1;
1, 3, 2;
1, 6, 10, 6;
1, 10, 30, 42, 23;
1, 15, 70, 168, 207, 106;
1, 21, 140, 504, 1035, 1166, 567;
...
Row 4 = (1, 10, 30, 42, 23) = termwise products of (1, 10, 15, 7, 1) and (1, 1, 2, 6, 23) = (1*1, 10*1, 15*2, 7*6, 1*23); where (1, 10, 15, 7, 1) = row 4 of triangle A085478. Q

Cf. A125273, A085478.

Triangle read by rows, T(n,k) = A085478(n,k) * A125273(k).

As infinite lower triangular matrices, A144250 = A085478 * (A125275 * 0^(n-k); where (A125275 * 0^(n-k)) = an infinite lower triangular matrix with A125275: (1, 1, 2, 6, 23, 106, 567, 3434,...) as the main diagonal and the rest zeros.

Corrected definition: Eigentriangle, row sums = A125273, shifted. - Gary W. Adamson, Nov 05 2008

A125276 Eigensequence of triangle A039598: a(n) = Sum_{k=0..n-1} A039598(n-1,k)*a(k) for n>0 with a(0)=1.

Original entry on oeis.org

1, 1, 3, 12, 58, 325, 2060, 14514, 112170, 941128, 8502393, 82160481, 844532873, 9191329357, 105491177081, 1272418794619, 16080824798705, 212370154398094, 2923859710010527, 41877072960374478, 622763691600244335

Offset: 0

Paul D. Hanna, Nov 26 2006

nonn

			a(3) = 5*(1) + 4*(1) + 1*(3) = 12;
a(4) = 14*(1) + 14*(1) + 6*(3) + 1*(12) = 58;
a(5) = 42*(1) + 48*(1) + 27*(3) + 8*(12) + 1*(58) = 325.
Triangle A039598(n,k) = C(2*n+2,n-k)*(k+1)/(n+1) begins:
1;
2, 1;
5, 4, 1;
14, 14, 6, 1;
42, 48, 27, 8, 1;
132, 165, 110, 44, 10, 1; ...
where g.f. of column k = G000108(x)^(2*k+2)
and G000108(x) = (1 - sqrt(1-4*x))/(2x) is the Catalan function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A039598, A000108; A125275 (variant).

A125276=ConstantArray[0,20]; A125276[[1]]=1; Do[A125276[[n]]=Binomial[2*n,n-1]/n+Sum[A125276[[k]]*Binomial[2*n,n-k-1]*(k+1)/n,{k,1,n-1}];,{n,2,20}]; Flatten[{1,A125276}] (* Vaclav Kotesovec, Dec 09 2013 *)

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

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

G.f. A(x) satisfies: A(x/(1+x)^2) = 1 + x*A(x); also, A(x*(1-x)) = 1 + [x/(1-x)]*A(x/(1-x)); also, A(x) = 1 + x*C(x)^2*A(x*C(x)^2) where C(x) = (1 - sqrt(1-4x))/(2x) is the Catalan function (A000108). - Paul D. Hanna, Aug 15 2007

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A147294 Eigentriangle, row sums = A125275.

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A144250 Eigentriangle, row sums = A125275, shifted.

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A125276 Eigensequence of triangle A039598: a(n) = Sum_{k=0..n-1} A039598(n-1,k)*a(k) for n>0 with a(0)=1.

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