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.

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

Offset

0,2

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Example

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, ...].

Prog

(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

Crossrefs

Cf. A323694, A295766.

Sequence in context: A105749 A251692 A338187 * A118086 A048163 A093548

Adjacent sequences: A323690 A323691 A323692 * A323694 A323695 A323696

Keyword

nonn

Author

Paul D. Hanna, Feb 20 2019

Status

approved