A118022 -id:A118022 - cp's OEIS frontend

A118023 Column 0 of triangle A118022, where the matrix square of A118022 shifts each column up 1 row, dropping the main diagonal of powers of 2.

1, 1, 3, 19, 243, 6227, 319251, 32737427, 6714170259, 2754046149011, 2259333156408723, 3706972573115098515, 12164337831474297132435, 79833941280970262512121235, 1047892334589811621056371520915

Offset: 0

Paul D. Hanna, Apr 10 2006

nonn

Numerators of the q-Catalan numbers for q = 1/2. - John Keith, Feb 19 2021

			1 = (1-x) + 1*x*(1-x)*(1-2*x) + 3*x^2*(1-x)*(1-2*x)*(1-4*x) + 19*x^3*(1-x)*(1-2*x)*(1-4*x)*(1-8*x) + 243*x^4*(1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x) + ...

Cf. A118022.

{a(n)=if(n==0, 1, polcoeff(1-sum(k=0, n-1, a(k)*x^k*prod(j=0, k, 1-2^j*x+x*O(x^n))), n))}
{a(n)=local(CF=1+x*O(x^n)); for(k=1, n, CF=1/(1-x/2^(n-k+1)*CF)); 2^(n*(n+1)/2)*polcoeff(CF, n)} \\ Paul D. Hanna, Sep 28 2012

G.f.: 1 = Sum_{n>=0} a(n)*x^n*prod_{k=0, n} (1-2^k*x) with a(0)=1.
a(n) = 2^(n*(n-1)/2)*b(n) where b(0)=1 and b(n)=sum(i=0,n-1,b(i)*b(n-1-i)/2^i). - Benoit Cloitre, Oct 25 2006
G.f.: Sum_{n>=0} a(n)*x^n/2^(n*(n+1)/2) = 1/(1 - (x/2)/(1 - (x/2^2)/(1 - (x/2^3)/(1 - (x/2^4)/(1 - (x/2^5)/(1 - ...)))))), a continued fraction. - Paul D. Hanna, Sep 28 2012

A118024 Triangle T, read by rows, T(n,k) = T(n-k)2^(k(n-k)) such that column 0 of the matrix square of T equals column 0 of T shifted left: [T^2](n,k) = T(n-k+1,0)2^(k(n-k)) for n>=k>=0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 8, 4, 1, 28, 48, 32, 8, 1, 216, 448, 384, 128, 16, 1, 2864, 6912, 7168, 3072, 512, 32, 1, 66656, 183296, 221184, 114688, 24576, 2048, 64, 1, 2760896, 8531968, 11730944, 7077888, 1835008, 196608, 8192, 128, 1, 205824384, 706789376

Offset: 0

Paul D. Hanna, Apr 10 2006

Column 0 is A118025, where T(n,k) = A118025(n-k)*2^(k*(n-k)).

			Triangle T begins:
1;
1,1;
2,2,1;
6,8,4,1;
28,48,32,8,1;
216,448,384,128,16,1;
2864,6912,7168,3072,512,32,1;
66656,183296,221184,114688,24576,2048,64,1; ...
2760896,8531968,11730944,7077888,1835008,196608,8192,128,1; ...
Matrix square is given by [T^2](n,k) = T(n-k+1,0)*2^(k*(n-k)):
1;
2,1;
6,4,1;
28,24,8,1;
216,224,96,16,1;
2864,3456,1792,384,32,1; ...
so that column 0 of T^2 equals column 0 of T shift left 1 place.

Cf. A118025 (column 0); A117401 (related triangle); A118022 (variant).

Cf. A123305.

{T(n, k)=if(n<0 || k>n,0,if(n==k,1,2^k*sum(j=0, n-1, T(n-1, j)*T(j, k)); ))} \\ Paul D. Hanna, Sep 25 2006

T(n,k) = A118025(n-k)*2^(k*(n-k)) for n>=k>=0.

cp's OEIS Frontend

A118023 Column 0 of triangle A118022, where the matrix square of A118022 shifts each column up 1 row, dropping the main diagonal of powers of 2.

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A118024 Triangle T, read by rows, T(n,k) = T(n-k)2^(k(n-k)) such that column 0 of the matrix square of T equals column 0 of T shifted left: [T^2](n,k) = T(n-k+1,0)2^(k(n-k)) for n>=k>=0.

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cp's OEIS Frontend

A118023 Column 0 of triangle A118022, where the matrix square of A118022 shifts each column up 1 row, dropping the main diagonal of powers of 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A118024 Triangle T, read by rows, T(n,k) = T(n-k)*2^(k*(n-k)) such that column 0 of the matrix square of T equals column 0 of T shifted left: [T^2](n,k) = T(n-k+1,0)*2^(k*(n-k)) for n>=k>=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A118024 Triangle T, read by rows, T(n,k) = T(n-k)2^(k(n-k)) such that column 0 of the matrix square of T equals column 0 of T shifted left: [T^2](n,k) = T(n-k+1,0)2^(k(n-k)) for n>=k>=0.