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 11-16 of 16 results.

A161552 E.g.f. satisfies: A(x,y) = exp(x*y*exp(x*A(x,y))).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 12, 1, 0, 4, 72, 48, 1, 0, 5, 320, 810, 160, 1, 0, 6, 1200, 8640, 6480, 480, 1, 0, 7, 4032, 70875, 143360, 42525, 1344, 1, 0, 8, 12544, 489888, 2240000, 1792000, 244944, 3584, 1, 0, 9, 36864, 3000564, 27869184, 49218750, 18579456, 1285956, 9216, 1
Offset: 0

Views

Author

Paul D. Hanna, Jun 13 2009, Jun 14 2009

Keywords

Comments

E.g.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n!.
Row sums, (n+1)^(n-1), equal A000272 (number of trees on n labeled nodes).

Examples

			Triangle begins:
1;
0,1;
0,2,1;
0,3,12,1;
0,4,72,48,1;
0,5,320,810,160,1;
0,6,1200,8640,6480,480,1;
0,7,4032,70875,143360,42525,1344,1;
0,8,12544,489888,2240000,1792000,244944,3584,1;
0,9,36864,3000564,27869184,49218750,18579456,1285956,9216,1; ...

Links

Crossrefs

Cf. A000272, A072590; A161565, A161566, A161567, A141369.

Programs

  • Mathematica
    Join[{1}, Table[Binomial[n, k]*(n - k + 1)^(k - 1)*k^(n - k), {n, 1, 10}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 18 2017 *)
  • PARI
    {T(n,k)=binomial(n,k)*(n-k+1)^(k-1)*k^(n-k)}
  • PARI
    {T(n,k)=local(A=1+x); for(i=0,n, A=exp(x*y*exp(x*A+O(x^n)))); n!*polcoeff(polcoeff(A,n,x),k,y)}

Formula

T(n,k) = binomial(n,k) * (n-k+1)^(k-1) * k^(n-k).
E.g.f. A(x,y) at y=1: A(x,1) = LambertW(-x)/(-x).
From Paul D. Hanna, Jun 14 2009: (Start)
More generally, if G(x) = exp(p*x*exp(q*x*G(x))),
where G(x)^m = Sum_{n>=0} g(n,m)*x^n/n!,
then g(n,m) = Sum_{k=0..n} C(n,k)*p^k*q^(n-k)*m*(n-k+m)^(k-1)*k^(n-k).
(End)

A367721 E.g.f. satisfies A(x) = exp(x*A(-x^2)).

Original entry on oeis.org

1, 1, 1, -5, -23, 1, 601, 7771, 26545, -401183, -6965999, -42828389, 528611161, 15543020065, 141983039017, -2393449681349, -83586615493919, -708151768946879, 15447932991283105, 635290179334026427, 7146984268771158601, -162583738763505944639
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2023

Keywords

Crossrefs

Cf. A141369, A367722, A367723.
Cf. A138292.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\2, (-1)^j*(2*j+1)*v[j+1]*v[i-2*j]/(j!*(i-1-2*j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * a(k) * a(n-1-2*k) / (k! * (n-1-2*k)!).

A367722 E.g.f. satisfies A(x) = exp(x*A(-x^3)).

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, 1681, 38641, 269137, 599761, -22461119, -347288039, -8704873319, -73184815703, 16491842641, 26323288948321, 725566429691041, 7867441656997921, -20568394299884543, -4768992217846599479, -108339469662214468439
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2023

Keywords

Comments

This sequence is different from A358063.

Crossrefs

Cf. A141369, A367721, A367723.
Cf. A358063.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\3, (-1)^j*(3*j+1)*v[j+1]*v[i-3*j]/(j!*(i-1-3*j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/3)} (-1)^k * (3*k+1) * a(k) * a(n-1-3*k) / (k! * (n-1-3*k)!).

A367723 E.g.f. satisfies A(x) = exp(x*A(-x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, -119, -719, -2519, -6719, 166321, 3598561, 29882161, 159572161, -389343239, -55939643759, -974399385959, -9282412863359, -46891283580959, 1814094098389441, 67045782535457761, 1076141148146824321, 61735522719009663721, 1058382395842664859121
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2023

Keywords

Crossrefs

Cf. A141369, A367721, A367722.
Cf. A367720.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\4, (-1)^j*(4*j+1)*v[j+1]*v[i-4*j]/(j!*(i-1-4*j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/4)} (-1)^k * (4*k+1) * a(k) * a(n-1-4*k) / (k! * (n-1-4*k)!).

A385140 E.g.f. A(x) satisfies A(x) = exp(2*x*A(-x)^(1/2)).

Original entry on oeis.org

1, 2, 0, -22, -16, 1042, 1792, -116758, -330496, 24101090, 96518144, -7976308118, -41609056256, 3875582805746, 25008143335424, -2601876338050582, -20048671462064128, 2308957345471798978, 20711293319504723968, -2618684079639256157974, -26823633677081126109184
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2025

Keywords

Crossrefs

Cf. A141369, A385141.
Cf. A360987.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A141369.
a(n) = 2 * Sum_{k=0..n} (n-k+2)^(k-1) * (-k)^(n-k) * binomial(n,k).

A385141 E.g.f. A(x) satisfies A(x) = exp(3*x*A(-x)^(1/3)).

Original entry on oeis.org

1, 3, 3, -36, -147, 1728, 14391, -193344, -2572263, 39702528, 744878859, -13061956608, -320684319675, 6310454624256, 192965057926335, -4214431981191168, -155017339047231951, 3722456794316931072, 160513751565607780755, -4204149732317088448512
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2025

Keywords

Crossrefs

Cf. A141369, A385140.
Cf. A360988.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A141369.
a(n) = 3 * Sum_{k=0..n} (n-k+3)^(k-1) * (-k)^(n-k) * binomial(n,k).
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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