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
Keywords
Examples
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.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..260
Programs
-
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
Formula
a(A075427(k) - 1) is odd for n>=0 and a(n) is even elsewhere (conjecture).
Comments