A129334
Triangle T(n,k) read by rows: inverse of the matrix PE = exp(P)/exp(1) given in A011971.
Original entry on oeis.org
1, -1, 1, 0, -2, 1, 1, 0, -3, 1, 1, 4, 0, -4, 1, -2, 5, 10, 0, -5, 1, -9, -12, 15, 20, 0, -6, 1, -9, -63, -42, 35, 35, 0, -7, 1, 50, -72, -252, -112, 70, 56, 0, -8, 1, 267, 450, -324, -756, -252, 126, 84, 0, -9, 1, 413, 2670, 2250, -1080, -1890, -504, 210, 120, 0, -10, 1
Offset: 0
Triangle starts:
[0] 1;
[1] -1, 1;
[2] 0, -2, 1;
[3] 1, 0, -3, 1;
[4] 1, 4, 0, -4, 1;
[5] -2, 5, 10, 0, -5, 1;
[6] -9, -12, 15, 20, 0, -6, 1;
[7] -9, -63, -42, 35, 35, 0, -7, 1;
[8] 50, -72, -252, -112, 70, 56, 0, -8, 1;
[9] 267, 450, -324, -756, -252, 126, 84, 0, -9, 1;
First column is
A000587 (Uppuluri Carpenter numbers) which is also the negative of the row sums (=P(n, 1)). Polynomials evaluated at 2 are
A074051, at -1
A109747.
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P := proc(n,x) option remember; if n=0 then 1 else
x*P(n-1, x) - P(n-1, x+1) fi end:
aRow := n -> seq(coeff(P(n, x), x, k), k = 0..n):
seq(aRow(n), n = 0..10); # Peter Luschny, Apr 15 2022
A367818
Expansion of e.g.f. exp(1 - 3*x - exp(x)).
Original entry on oeis.org
1, -4, 15, -53, 178, -575, 1809, -5598, 17141, -52113, 157724, -475997, 1433429, -4311364, 12958627, -38909601, 116831426, -350844883, 1051414421, -3160120038, 9491592177, -28218244109, 86403627444, -255153772169, 722619907385, -2772952748516, 4627276967623, -17420488524253
Offset: 0
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nmax = 27; CoefficientList[Series[Exp[1 - 3 x - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, -1], {k, 0, n}], {n, 0, 27}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 3*x - exp(x)))) \\ Michel Marcus, Dec 02 2023
A367819
Expansion of e.g.f. exp(1 - 4*x - exp(x)).
Original entry on oeis.org
1, -5, 24, -111, 497, -2166, 9239, -38765, 160658, -659773, 2691205, -10922544, 44166173, -178098121, 716703848, -2879774019, 11558005677, -46348854134, 185746261419, -744036460097, 2979305960426, -11926715433881, 47735079979633, -191026723545976, 764362047956073, -3058170811731677
Offset: 0
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nmax = 25; CoefficientList[Series[Exp[1 - 4 x - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -4 a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, -1], {k, 0, n}], {n, 0, 25}]
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my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 4*x - exp(x)))) \\ Michel Marcus, Dec 02 2023
A367784
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (n*k - 1)^n / k!.
Original entry on oeis.org
1, -2, 5, 17, 17, -8151, -311435, -777974, 927723585, 82906687673, 1693962380101, -707005824990631, -137258747025993071, -10253960705018807830, 1697644859939460151413, 803696888217607331079149, 148126297324647875348070657, -323461353221296480463456191
Offset: 0
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Table[n! SeriesCoefficient[Exp[1 - x - Exp[n x]], {x, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[Sum[(-1)^(n - k) Binomial[n, k] n^k BellB[k, -1], {k, 0, n}], {n, 0, 17}]
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