A295766 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2)/n^2 = [x^(n-2)] A(x)^(n^2) for n>=2 with A'(0) = 1.

1, 1, 5, 90, 3204, 170987, 12162683, 1087504130, 118227836360, 15304211345298, 2324856843115770, 409872125913866852, 83092182794794380856, 19214014336799266619671, 5030971580159960051721815, 1481724835890098667273954338, 487883202104697456579537247232, 178595806151469762148235569612814, 72312528698655521190143801630975174

Offset

0,3

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..260

Formula

a(A075427(k) - 1) is odd for n>=0 and a(n) is even elsewhere (conjecture).

Example

G.f.: A(x) = 1 + x + 5*x^2 + 90*x^3 + 3204*x^4 + 170987*x^5 + 12162683*x^6 + 1087504130*x^7 + 118227836360*x^8 + 15304211345298*x^9 + 2324856843115770*x^10 + ...
ILLUSTRATION OF THE DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [1, 1, 5, 90, 3204, 170987, 12162683, ...];
n=2: [1, 4, 26, 424, 14107, 729196, 50993674, ...];
n=3: [1, 9, 81, 1254, 37602, 1833597, 124332453, ...];
n=4: [1, 16, 200, 3200, 86084, 3846720, 248466736, ...];
n=5: [1, 25, 425, 7550, 188750, 7566705, 455263225, ...];
n=6: [1, 36, 810, 16680, 410499, 14777964, 808802730, ...];
n=7: [1, 49, 1421, 34594, 886312, 29473255, 1444189495, ...]; ...
in which the main diagonal
[1, 4, 81, 3200, 188750, 14777964, 1444189495, ...]
is related to an adjacent diagonal by dividing by n^2 like so:
[1, 4/4, 81/9, 3200/16, 188750/25, 14777964/36, 1444189495/49, ...]
= [1, 1, 9, 200, 7550, 410499, 29473255, ...].
Thus [x^(n-1)] A(x)^(n^2)/n^2 = [x^(n-2)] A(x)^(n^2) for n>=2.

Prog

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

Crossrefs

Cf. A088715, A295811, A075427.

Sequence in context: A277303 A361551 A323570 * A361586 A367500 A216693

Adjacent sequences: A295763 A295764 A295765 * A295767 A295768 A295769

Keyword

nonn

Author

Paul D. Hanna, Jan 31 2018

Status

approved