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.

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A295811 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = 2*n * [x^(n-2)] A(x)^(n^2) for n>=2, with A(0) = 1.

Original entry on oeis.org

1, 1, 2, 11, 140, 2898, 80844, 2786091, 113184008, 5266198778, 275248731860, 15939117549502, 1012084698990904, 69901132180300132, 5217426460077854712, 418615099531669351443, 35942031310982080239120, 3289533291926922095871546, 319841125714352173292953668, 32937612567848507536114539402, 3582858531960091228861488651864
Offset: 0

Views

Author

Paul D. Hanna, Feb 02 2018

Keywords

Comments

Compare g.f. to: [x^(n-1)] G(x)^n = 2 * [x^(n-2)] G(x)^n for n>=2 holds when G(x) = 1/(1-x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 140*x^4 + 2898*x^5 + 80844*x^6 + 2786091*x^7 + 113184008*x^8 + 5266198778*x^9 + 275248731860*x^10 + ...
ILLUSTRATION OF THE DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [1, 1, 2, 11, 140, 2898, 80844, ...];
n=2: [1, 4, 14, 72, 741, 13724, 364546, ...];
n=3: [1, 9, 54, 327, 2826, 42660, 1017720, ...];
n=4: [1, 16, 152, 1216, 10540, 129376, 2559792, ...];
n=5: [1, 25, 350, 3775, 37750, 427480, 6820800, ...];
n=6: [1, 36, 702, 10056, 123165, 1477980, 20712546, ...];
n=7: [1, 49, 1274, 23667, 359856, 4953998, 69355972, ...]; ...
in which the main diagonal
D0 = [1, 4, 54, 1216, 37750, 1477980, 69355972, 3775816704, ...]
and the adjacent diagonal
D1 = [1, 9, 152, 3775, 123165, 4953998, 235988544, 12954335103, ...]
are related by D0[n-1] = 2*n*D1[n-2] for n>=2.
The related sequence D0[n-1]/n^2, n>=1, begins:
[1, 1, 6, 76, 1510, 41055, 1415428, 58997136, 2878741134, 160698224230, ...].

Links

Crossrefs

Cf. A295766, A088715, A300873.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=2, n+1, A=concat(A, 0); V=Vec(Ser(A)^(m^2)); A[#A] = V[#A-1]*2/m - V[#A]/m^2 ); A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

a(2^k - 1) is odd for k>=0 and a(n) is even elsewhere (conjecture).
a(n) ~ c * d^n * n! / n^3, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.176554609483480358231680164050876553672889794284... and c = 2.719099850893334482... - Vaclav Kotesovec, Feb 07 2018

A323693 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = (n+1)^2 * [x^(n-1)] A(x)^(n+1) for n >= 1 with A'(0) = 1.

Original entry on oeis.org

1, 2, 14, 228, 6332, 255800, 13862744, 962576816, 83146713104, 8746885895136, 1102050352603232, 163997224386523712, 28480503345597714112, 5711832009893579651456, 1310680283957123653000064, 341305200596595166803458816, 100122955976950431349888239872, 32871729257928892872345863470592, 12007438407819424861612909690881536, 4854069613493626427129286480218215424
Offset: 0

Views

Author

Paul D. Hanna, Feb 20 2019

Keywords

Comments

a(n) / 2^floor((n+1)/2) is odd for n >= 0 (conjecture).

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 228*x^3 + 6332*x^4 + 255800*x^5 + 13862744*x^6 + 962576816*x^7 + 83146713104*x^8 + 8746885895136*x^9 + ...
The table of coefficients of x^k in A(x)^n starts as
n=1: [1, 2, 14, 228, 6332, 255800, 13862744, ...];
n=2: [1, 4, 32, 512, 13772, 543312, 28977968, ...];
n=3: [1, 6, 54, 860, 22488, 866448, 45462704, ...];
n=4: [1, 8, 80, 1280, 32664, 1229568, 63445984, ...];
n=5: [1, 10, 110, 1780, 44500, 1637512, 83069960, ...];
n=6: [1, 12, 144, 2368, 58212, 2095632, 104491088, ...];
n=7: [1, 14, 182, 3052, 74032, 2609824, 127881376, ...]; ...
RELATED SEQUENCES.
In the above table, the main diagonal begins
[1, 4, 54, 1280, 44500, 2095632, 127881376, 9819500544, ...]
which, when divided by (n+1)^2, yields the secondary diagonal (A323694):
[1, 1, 6, 80, 1780, 58212, 2609824, 153429696, 11457990000, ...].
The sequence a(n) / 2^floor((n+1)/2) appears to consist only of odd numbers:
[1, 1, 7, 57, 1583, 31975, 1732843, 60161051, 5196669569, 273340184223, 34439073518851, 2562456631039433, 445007864774964283, ...].

Links

Crossrefs

Cf. A323694, A295766.

Programs

  • PARI
    {a(n) = my(A=[1], V); for(m=2, n+1, A=concat(A, 0); V=Vec(Ser(A)^m); A[#A] = V[#A-1]*m - V[#A]/m ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Informal method of obtaining N terms: */
N=30; A=[1]; for(n=2, N, A=concat(A, 0); V=Vec(Ser(A)^n); A[#A] = V[#A-1]*n - V[#A]/n ); A
Showing 1-2 of 2 results.

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