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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. 2

%I #10 Feb 20 2019 12:38:17

%S 1,2,14,228,6332,255800,13862744,962576816,83146713104,8746885895136,

%T 1102050352603232,163997224386523712,28480503345597714112,

%U 5711832009893579651456,1310680283957123653000064,341305200596595166803458816,100122955976950431349888239872,32871729257928892872345863470592,12007438407819424861612909690881536,4854069613493626427129286480218215424

%N 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.

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

%H Paul D. Hanna, <a href="/A323693/b323693.txt">Table of n, a(n) for n = 0..300</a>

%e 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 + ...

%e The table of coefficients of x^k in A(x)^n starts as

%e n=1: [1, 2, 14, 228, 6332, 255800, 13862744, ...];

%e n=2: [1, 4, 32, 512, 13772, 543312, 28977968, ...];

%e n=3: [1, 6, 54, 860, 22488, 866448, 45462704, ...];

%e n=4: [1, 8, 80, 1280, 32664, 1229568, 63445984, ...];

%e n=5: [1, 10, 110, 1780, 44500, 1637512, 83069960, ...];

%e n=6: [1, 12, 144, 2368, 58212, 2095632, 104491088, ...];

%e n=7: [1, 14, 182, 3052, 74032, 2609824, 127881376, ...]; ...

%e RELATED SEQUENCES.

%e In the above table, the main diagonal begins

%e [1, 4, 54, 1280, 44500, 2095632, 127881376, 9819500544, ...]

%e which, when divided by (n+1)^2, yields the secondary diagonal (A323694):

%e [1, 1, 6, 80, 1780, 58212, 2609824, 153429696, 11457990000, ...].

%e The sequence a(n) / 2^floor((n+1)/2) appears to consist only of odd numbers:

%e [1, 1, 7, 57, 1583, 31975, 1732843, 60161051, 5196669569, 273340184223, 34439073518851, 2562456631039433, 445007864774964283, ...].

%o (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]}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) /* Informal method of obtaining N terms: */

%o 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

%Y Cf. A323694, A295766.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 20 2019

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)