A320079
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 14, 0, 1, 4, 21, 76, 88, 0, 1, 5, 36, 222, 772, 694, 0, 1, 6, 55, 488, 3132, 9808, 6578, 0, 1, 7, 78, 910, 8824, 55242, 149552, 72792, 0, 1, 8, 105, 1524, 20080, 199456, 1169262, 2660544, 920904, 0, 1, 9, 136, 2366, 39708, 553870, 5410208, 28873800, 54093696, 13109088, 0
Offset: 0
E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k + 1)*x^2/2! + 2*k*(3*k^2 + 3*k + 1)*x^3/3! + 2*k*(12*k^3 + 18*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 3, 10, 21, 36, 55, ...
0, 14, 76, 222, 488, 910, ...
0, 88, 772, 3132, 8824, 20080, ...
0, 694, 9808, 55242, 199456, 553870, ...
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Table[Function[k, n! SeriesCoefficient[1/(1 + k Log[1 - x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
A354147
Expansion of e.g.f. 1/(1 - 4 * log(1+x)).
Original entry on oeis.org
1, 4, 28, 296, 4168, 73376, 1550048, 38202048, 1076017344, 34096092672, 1200459182592, 46492497859584, 1964295942558720, 89906908894150656, 4431634108980264960, 234044235939806232576, 13184410813249253031936, 789137065405617987354624
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-4*log(1+x))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=4*sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;
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a(n) = sum(k=0, n, 4^k*k!*stirling(n, k, 1));
A354751
Expansion of e.g.f. 1 / (1 - log(1 + 4*x) / 4).
Original entry on oeis.org
1, 1, -2, 14, -152, 2264, -42832, 982512, -26484096, 820207488, -28692711168, 1118821622016, -48112717347840, 2261868010650624, -115400220781209600, 6350152838136428544, -374874781697133871104, 23632196147497381625856, -1584445791263626895228928
Offset: 0
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nmax = 18; CoefficientList[Series[1/(1 - Log[1 + 4 x]/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 4^(n - k), {k, 0, n}], {n, 0, 18}]
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my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+4*x)/4))) \\ Michel Marcus, Jun 06 2022
A365584
Expansion of e.g.f. 1 / (1 + 4 * log(1-x))^(3/4).
Original entry on oeis.org
1, 3, 24, 300, 5100, 109692, 2854344, 87164088, 3055516800, 120916282368, 5331444120576, 259168711406976, 13769882994784896, 793844510730348672, 49353915922852214016, 3291455140392403401984, 234388011123877880424960, 17750517946502792294592000
Offset: 0
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a[n_] := Sum[Product[4*j + 3, {j, 0, k - 1}] * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 10 2023 *)
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a(n) = sum(k=0, n, prod(j=0, k-1, 4*j+3)*abs(stirling(n, k, 1)));
Showing 1-4 of 4 results.