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.

A113664 Self-convolution 4th power of A113670, where a(n) = A113670(n+1)/(n+1).

Original entry on oeis.org

1, 4, 38, 556, 10745, 254700, 7100444, 226687964, 8138624340, 324257974800, 14191562670382, 676772204063828, 34931592000218062, 1940427169943222088, 115440543670528170360, 7323969420842077029820, 493653199877341017573868
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. satisfies: A(x) = (1 + x*d/dx[x*A(x)] )^4.
a(n) ~ c * 4^n * n! * n^(3/4), where c = 0.71282302257258522141135342... - Vaclav Kotesovec, Oct 23 2020

A113669 Self-convolution cube equals A113663, where a(n) = n*A113663(n-1) for n>=1, with a(0)=1.

Original entry on oeis.org

1, 1, 6, 63, 904, 16080, 337374, 8107743, 218940480, 6554205342, 215319184860, 7701064928370, 297912862462680, 12396725926132990, 552257670588677214, 26229243983909050215, 1323230977463353055616, 70673562984581535191094
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. A(x) satisfies:
(1) A(x) = 1 + x*d/dx[x*A(x)^3],
(2) [x^n] exp( x*A(x)^3 ) * (n + 1 - A(x)) = 0 for n > 0,
(3) [x^n] exp( n * x*A(x)^3 ) * (2 - A(x)) = 0 for n > 0. - Paul D. Hanna, May 27 2018
From Vaclav Kotesovec, Oct 23 2020: (Start)
a(n) ~ c * 3^n * n! * n^(2/3), where c = 0.2509528330393045762351289...
a(n) ~ A113663(n)/3. (End)
a(0) = 1; a(n) = n * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Jul 25 2021

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.