cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A113663 Self-convolution cube of A113669, where a(n) = A113669(n+1)/(n+1).

Original entry on oeis.org

1, 3, 21, 226, 3216, 56229, 1158249, 27367560, 728245038, 21531918486, 700096811670, 24826071871890, 953594302010230, 39446976470619801, 1748616265593936681, 82701936091459565976, 4157268410857737364182
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2005

Keywords

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n, A=(1+x*deriv(x*A))^3);polcoeff(A,n,x)}

Formula

G.f. satisfies: A(x) = (1 + x*d/dx[x*A(x)] )^3.
a(n) ~ c * 3^n * n! * n^(2/3), where c = 0.7528584991179137287053869... - Vaclav Kotesovec, Oct 23 2020

A113670 Self-convolution 4th power equals A113664, where a(n) = n*A113664(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 8, 114, 2224, 53725, 1528200, 49703108, 1813503712, 73247619060, 3242579748000, 156107189374202, 8121266448765936, 454110696002834806, 27165980379205109232, 1731608155057922555400, 117183510733473232477120
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2005

Keywords

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^4));polcoeff(A,n,x)}

Formula

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^4],
(2) [x^n] exp( x*A(x)^4 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^4 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018

A113671 Self-convolution 5th power equals A113665, where a(n) = n*A113665(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 10, 180, 4440, 135525, 4866156, 199577910, 9174096960, 466435229220, 25973117225450, 1571873641094680, 102741164109622800, 7214517196021315830, 541781124945022815720, 43336510897320779553450
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2005

Keywords

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^5));polcoeff(A,n,x)}

Formula

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^5],
(2) [x^n] exp( x*A(x)^5 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^5 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018

A113672 Self-convolution 6th power equals A113666, where a(n) = n*A113666(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 12, 261, 7784, 287145, 12452256, 616408534, 34178166288, 2094929612766, 140568321437700, 10246761825942972, 806426083421461440, 68162575162983744079, 6159817390723312545936, 592796927295190983761100
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2005

Keywords

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^6));polcoeff(A,n,x)}

Formula

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^6],
(2) [x^n] exp( x*A(x)^6 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^6 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018

A113673 Self-convolution 7th power equals A113667, where a(n) = n*A113667(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 14, 357, 12488, 540155, 27453258, 1591997162, 103362754048, 7415833578300, 582246803894350, 49648781879763836, 4569614321483063496, 451606519694514555917, 47709061981854231868308
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2005

Keywords

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^7));polcoeff(A,n,x)}

Formula

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^7],
(2) [x^n] exp( x*A(x)^7 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^7 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018

A113674 Self-convolution 8th power equals A113668, where a(n) = n*A113668(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 16, 468, 18784, 932030, 54321840, 3611129620, 268687287744, 22085224470873, 1986091468594160, 193935237759263880, 20436302307290415264, 2311999369405933686648, 279558778132903394262032
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2005

Keywords

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^8));polcoeff(A,n,x)}

Formula

G.f. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^8],
(2) [x^n] exp( x*A(x)^8 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^8 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018
Showing 1-6 of 6 results.