A323694 a(n) = [x^n] G(x)^(n+1) / (n+1)^2 for n >= 0, where G(x) is the g.f. of A323693.
1, 1, 6, 80, 1780, 58212, 2609824, 153429696, 11457990000, 1060569950000, 119272908950624, 16028231512305792, 2537651853233579264, 467637448273232473600, 99254069921234336025600, 24041252393859722676492288, 6592065384977916153080977152, 2031547109837445197745286936320, 699189635655553248110260224524800
Offset: 0
Keywords
Examples
The g.f. of A323693 begins G(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 G(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^2, yields this sequence: [1, 1, 6, 80, 1780, 58212, 2609824, 153429696, 11457990000, ...] and also yields the secondary diagonal in the above table.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Crossrefs
Cf. A323693.
Programs
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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 ); polcoeff( Ser(A)^(n+1),n)/(n+1)^2} for(n=0, 20, print1(a(n), ", "))
Formula
a(n) = [x^(n-1)] G(x)^(n+1) for n >= 1, where G(x) is the g.f. of A323693.
Comments