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.

Previous Showing 31-34 of 34 results.

A380808 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x) / (1 + x*exp(-x)) ).

Original entry on oeis.org

1, 3, 24, 335, 6812, 183397, 6168406, 249350285, 11785793352, 638146503593, 38960123581154, 2648475653518081, 198429466488527164, 16246940820392924189, 1443430758561178861758, 138305198841617791230533, 14217431594874334746229520, 1560842183273111251153540945
Offset: 0

Views

Author

Seiichi Manyama, Feb 04 2025

Keywords

Links

Crossrefs

Cf. A088690, A161633, A380826.
Cf. A352448, A380828.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+k+2)^k*binomial(n, k)/(k+1)!);

Formula

E.g.f. A(x) satisfies A(x) = exp(2*x*A(x)) / ( 1 - x*exp(x*A(x)) ).
a(n) = n! * Sum_{k=0..n} (n+k+2)^k * binomial(n,k)/(k+1)!.

A380080 Expansion of e.g.f. (1/x) * Series_Reversion( x / sqrt(1 + 2*x*exp(x)) ).

Original entry on oeis.org

1, 1, 3, 15, 109, 1045, 12501, 179599, 3015657, 57988809, 1257058585, 30337358491, 806837271021, 23448335293981, 739379851041573, 25143044445680295, 917252832237053521, 35735484803144976145, 1480838869407287923569, 65038486139094829172275, 3017945328547452509505045
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A161633, A380081.
Cf. A380050.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, 2^k*k^(n-k)*binomial(n/2+1/2, k)/(n-k)!)/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = sqrt( 1 + 2*x*A(x)*exp(x*A(x)) ).
a(n) = (n!/(n+1)) * Sum_{k=0..n} 2^k * k^(n-k) * binomial(n/2+1/2,k)/(n-k)!.

A380081 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + 3*x*exp(x))^(1/3) ).

Original entry on oeis.org

1, 1, 2, 7, 36, 245, 2086, 21357, 255704, 3507625, 54258570, 934600601, 17743468612, 368146983789, 8288468950958, 201258635444245, 5243025162331056, 145871455305823697, 4316920830720239122, 135408946029576741297, 4487574630295937337500, 156686063319198543135061
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A161633, A380080.
Cf. A380051.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, 3^k*k^(n-k)*binomial(n/3+1/3, k)/(n-k)!)/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = ( 1 + 3*x*A(x)*exp(x*A(x)) )^(1/3).
a(n) = (n!/(n+1)) * Sum_{k=0..n} 3^k * k^(n-k) * binomial(n/3+1/3,k)/(n-k)!.

A366230 Expansion of e.g.f. A(x,y) satisfying A(x,y) = 1 + x*A(x,y) * exp(x*y * A(x,y)), as a triangle read by rows.

Original entry on oeis.org

1, 1, 0, 2, 2, 0, 6, 18, 3, 0, 24, 144, 96, 4, 0, 120, 1200, 1800, 400, 5, 0, 720, 10800, 28800, 16200, 1440, 6, 0, 5040, 105840, 441000, 470400, 119070, 4704, 7, 0, 40320, 1128960, 6773760, 11760000, 6021120, 762048, 14336, 8, 0, 362880, 13063680, 106686720, 274337280, 238140000, 65028096, 4408992, 41472, 9, 0
Offset: 0

Views

Author

Paul D. Hanna, Nov 17 2023

Keywords

Comments

A161633(n) = Sum_{k=0..n} T(n,k) for n >= 0.
A366232(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.
A366233(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.
A366234(n) = Sum_{k=0..n} T(n,k) * 4^k for n >= 0.
A366235(n) = Sum_{k=0..n} T(n,k) * 5^k for n >= 0.

Examples

			E.g.f. A(x,y) = 1 + x + (2*y + 2)*x^2/2! + (3*y^2 + 18*y + 6)*x^3/3! + (4*y^3 + 96*y^2 + 144*y + 24)*x^4/4! + (5*y^4 + 400*y^3 + 1800*y^2 + 1200*y + 120)*x^5/5! + (6*y^5 + 1440*y^4 + 16200*y^3 + 28800*y^2 + 10800*y + 720)*x^6/6! + (7*y^6 + 4704*y^5 + 119070*y^4 + 470400*y^3 + 441000*y^2 + 105840*y + 5040)*x^7/7! + (8*y^7 + 14336*y^6 + 762048*y^5 + 6021120*y^4 + 11760000*y^3 + 6773760*y^2 + 1128960*y + 40320)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y) begins
 1;
 1, 0;
 2, 2, 0;
 6, 18, 3, 0;
 24, 144, 96, 4, 0;
 120, 1200, 1800, 400, 5, 0;
 720, 10800, 28800, 16200, 1440, 6, 0;
 5040, 105840, 441000, 470400, 119070, 4704, 7, 0;
 40320, 1128960, 6773760, 11760000, 6021120, 762048, 14336, 8, 0;
 362880, 13063680, 106686720, 274337280, 238140000, 65028096, 4408992, 41472, 9, 0;
 ...

Crossrefs

Cf. A161633, A366232, A366233, A366234, A366235.

Programs

  • PARI
    {T(n,k) = n! * binomial(n+1, n-k)/(n+1) * (n-k)^k/k!}
for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))

Formula

T(n,k) = n! * binomial(n+1, n-k)/(n+1) * (n-k)^k / k!.
Let A(x,y)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * y^k * (n-k)^k/k!.
E.g.f. A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} T(n,k)*y^k satisfies the following formulas.
(1) A(x,y) = 1 + x*A(x) * exp(x*y*A(x,y)).
(2) A(x,y) = (1/x) * Series_Reversion( x/(1 + x*exp(x*y)) ).
(3) A( x/(1 + x*exp(x*y)), y) = 1 + x*exp(x*y).
(4) A(x,y) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+m-y)*x*A(x,y)) for all fixed nonnegative m.
(4.a) A(x,y) = 1 + Sum{n>=1} n^(n-1) * x^n/n! * A(x,y)^n * exp(-(n-y)*x*A(x)).
(4.b) A(x,y) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+1-y)*x*A(x,y)).
(4.c) A(x,y) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+2-y)*x*A(x,y)).
(4.d) A(x,y) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+3-y)*x*A(x,y)).
(4.e) A(x,y) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+4-y)*x*A(x,y)).
Previous Showing 31-34 of 34 results.

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