A237652 G.f. satisfies: [x^n] A(x)^(n^2) = [x^n] A(x)^(n^2-1) for n>1 with A(0)=A'(0)=1.
1, 1, -3, 20, -245, 4290, -114422, 4086800, -203647509, 12920587070, -1053926397590, 105178069321944, -12765014959365682, 1838898931467398164, -311221726754896488780, 61047560951879121055296, -13747598006865584455353165, 3521759025274977423306328182, -1018406456608128511401443183654
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x - 3*x^2 + 20*x^3 - 245*x^4 + 4290*x^5 - 114422*x^6 +... The coefficients in relevant powers of g.f. A(x) begin: A^3: [1, 3, (-6), 43, -597, 11127, -313038, 11486268, ...]; A^4: [1, 4, (-6), 48, -721, 13836, -399342, 14835168, ...]; ... A^8: [1, 8, 4, (48), -1022, 21328, -677040, 26240352, ...]; A^9: [1, 9, 9, (48), -1071, 22572, -732768, 28655712, ...]; ... A^15: [1, 15, 60, 125, (-1260), 26508, -986720, 40214775, ...]; A^16: [1, 16, 72, 160, (-1260), 26688, -1018704, 41720576, ...]; ... A^24: [1, 24, 204, 848, 54, (25680), -1211936, 50397024, ...]; A^25: [1, 25, 225, 1000, 525, (25680), -1230900, 51117200, ...]; ... A^35: [1, 35, 490, 3675, 14035, 52927, (-1360590), 54736260, ...]; A^36: [1, 36, 522, 4080, 16695, 61452, (-1360590), 54781344, ...]; ... A^48: [1, 48, 984, 11488, 82428, 399936, -450096, (53190144), ...]; A^49: [1, 49, 1029, 12348, 91679, 460110, -217266, (53190144), ...]; ... which illustrates [x^n] A(x)^(n^2-1) = [x^n] A(x)^(n^2) for n>1.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..150
Programs
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PARI
{a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[ #A]=(Vec(Ser(A)^((#A-1)^2-1))-Vec(Ser(A)^((#A-1)^2)))[ #A]); A[n+1]} for(n=0,30,print1(a(n),", "))