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.

A113668 Self-convolution 8th power of A113674, where a(n) = A113674(n+1)/(n+1).

Original entry on oeis.org

1, 8, 156, 4696, 186406, 9053640, 515875660, 33585910968, 2453913830097, 198609146859416, 17630476159933080, 1703025192274201272, 177846105338917975896, 19968484152350242447288
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2005

Keywords

Comments

From Vaclav Kotesovec, Oct 23 2020: (Start)
In general, for k>=1, if g.f. satisfies: A(x) = (1 + x*d/dx[x*A(x)] )^k, then a(n) ~ c(k) * k^n * n! * n^((k-1)/k), where c(k) is a constant (dependent only on k).
c(k) tends to A238223*exp(1) = 0.592451670452494179138706... if k tends to infinity.
(End)

Links

Crossrefs

Cf. A113674, A113662, A113663, A113664, A113665, A113666, A113667, A338377.

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. satisfies: A(x) = (1 + x*d/dx[x*A(x)] )^8.
a(n) ~ c * 8^n * n! * n^(7/8), where c = 0.6523348263871879460325... - 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

Links

Crossrefs

Cf. A113663, A000699, A113670, A113671, A113672, A113673, A113674.

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

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

Cf. A113664, A000699, A113669, A113671, A113672, A113673, A113674.

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

Cf. A113665, A000699, A113669, A113670, A113672, A113673, A113674.

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

Cf. A113666, A000699, A113669, A113670, A113671, A113673, A113674.

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

Cf. A113667, A000699, A113669, A113670, A113671, A113672, A113674.

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
Showing 1-6 of 6 results.

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