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 A263075 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = n^n * (n-1)! for n>=1. 6
 1, 1, 2, 31, 1150, 68713, 5914776, 692005074, 105932315154, 20617891510063, 4984425649932314, 1467604324373250545, 517561005098562714944, 215501019188749426210440, 104642607303457024105207408, 58625315029802441203026824094, 37541542090285460025870424920666 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS It appears that for k>0, a(k) is odd iff k = 2*A003714(n)+1 for n>=0, where A003714 is the fibbinary numbers (integers whose binary representation contains no consecutive ones); this is true for at least the first 531 terms. [See also A263190 and A171791.] LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA a(n) ~ exp(1-exp(-1)) * n! * n^(n-1). - Vaclav Kotesovec, Oct 20 2020 EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 31*x^3 + 1150*x^4 + 68713*x^5 + 5914776*x^6 + 692005074*x^7 + 105932315154*x^8 +... The coefficients in A(x)^(n^2) begin: n=1: [1, 1, 2, 31, 1150, 68713, 5914776, 692005074, 105932315154, ...]; n=2: [1, 4, 14, 152, 5021, 289824, 24532494, 2841972672, 432284291486, ...]; n=3: [1, 9, 54, 507, 13356, 715635, 58722228, 6685822296, 1005887241243, ...]; n=4: [1, 16, 152, 1536, 31500, 1468016, 114260704, 12668897920, ...]; n=5: [1, 25, 350, 4275, 75000, 2840855, 202155100, 21547156900, ...]; n=6: [1, 36, 702, 10776, 184725, 5598720, 344795598, 34598389248, ...]; n=7: [1, 49, 1274, 24647, 456386, 11753973, 592950960, 54103596918, ...]; n=8: [1, 64, 2144, 51712, 1092016, 26366656, 1071635712, 84557168640, ...]; n=9: [1, 81, 3402, 100791, 2482650, 61309629, 2096140032, 135856780686, ...]; ... where the terms along the main diagonal begin: [1, 4, 54, 1536, 75000, 5598720, 592950960, 84557168640, ..., n^n*(n-1)!, ...]. Note that odd terms a(n) occur at positions n starting with: [0, 1, 3, 5, 9, 11, 17, 19, 21, 33, 35, 37, 41, 43, 65, 67, 69, 73, 75, 81, 83, 85, 129, 131, 133, 137, 139, 145, 147, 149, 161, 163, 165, 169, 171, 257, ...], which seems to equal A118113, the even Fibbinary numbers + 1, with an initial zero included. PROG (PARI) {a(n) = local(A=[1, 1]); for(i=1, n+1, A=concat(A, 0); m=#A; A[m] = ( m^m*(m-1)! - Vec(Ser(A)^(m^2))[m] )/m^2 ); A[n+1]} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A171791, A263190, A003714, A118113, A244589. Sequence in context: A333349 A349071 A224863 * A217766 A246970 A246969 Adjacent sequences: A263072 A263073 A263074 * A263076 A263077 A263078 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 08 2015 STATUS approved

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Last modified February 24 20:59 EST 2024. Contains 370307 sequences. (Running on oeis4.)