Search: a292861 -id:a292861
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A292860
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Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).
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11
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1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 5, 0, 1, 4, 12, 22, 15, 0, 1, 5, 20, 57, 94, 52, 0, 1, 6, 30, 116, 309, 454, 203, 0, 1, 7, 42, 205, 756, 1866, 2430, 877, 0, 1, 8, 56, 330, 1555, 5428, 12351, 14214, 4140, 0, 1, 9, 72, 497, 2850, 12880, 42356, 88563, 89918, 21147, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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A(0,k) = 1 and A(n,k) = k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 6, 12, 20, 30, 42, ...
0, 5, 22, 57, 116, 205, 330, ...
0, 15, 94, 309, 756, 1555, 2850, ...
0, 52, 454, 1866, 5428, 12880, 26682, ...
0, 203, 2430, 12351, 42356, 115155, 268098, ...
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MAPLE
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A:= proc(n, k) option remember; `if`(n=0, 1,
(1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
end:
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MATHEMATICA
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A[0, _] = 1; A[n_ /; n >= 0, k_ /; k >= 0] := A[n, k] = k*Sum[Binomial[n-1, j]*A[j, k], {j, 0, n-1}]; A[_, _] = 0;
A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263, A221159, A276506, A276507.
Same array, different indexing is A189233.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A292866
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a(n) = n! * [x^n] exp(n*(1 - exp(x))).
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10
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1, -1, 2, -3, -20, 370, -4074, 34293, -138312, -2932533, 106271090, -2192834490, 32208497124, -206343936097, -7657279887698, 412496622532785, -12455477719752976, 260294034150380430, -2256541295745391542, -122593550603339550843, 8728842979656718306780
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (-n)^k * Stirling2(n,k). - Seiichi Manyama, Jul 28 2019
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MAPLE
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b:= proc(n, k) option remember; `if`(n=0, 1,
-(1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
end:
a:= n-> b(n$2):
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MATHEMATICA
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Table[n!*SeriesCoefficient[E^(n*(1 - E^x)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 25 2017 *)
a[n_] := BellB[n, -n]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Dec 23 2021 *)
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PROG
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(Ruby)
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
ary = [1]
(1..n).each{|i| ary << k * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[j]}}
ary
end
(0..n).map{|i| A(-i, i)[-1]}
end
(PARI) {a(n) = sum(k=0, n, (-n)^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 28 2019
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A309084
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a(n) = exp(3) * Sum_{k>=0} (-3)^k*k^n/k!.
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4
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1, -3, 6, -3, -21, 24, 195, -111, -3072, -4053, 57003, 277854, -697539, -12261567, -29861778, 371727465, 3511027599, 2028432480, -188521156857, -1470389129931, 1655487186864, 121873222577823, 915525253963023, -2095901567014530, -103715912230195863, -836215492271268459
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Sum_{j>=0} (-3)^j*x^j / Product_{k=1..j} (1 - k*x).
E.g.f.: exp(3*(1 - exp(x))).
a(n) = Sum_{k=0..n} (-3)^k * Stirling2(n,k).
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MAPLE
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b:= proc(n, m) option remember; `if`(n=0,
(-3)^m, m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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Table[Exp[3] Sum[(-3)^k k^n/k!, {k, 0, Infinity}], {n, 0, 25}]
Table[BellB[n, -3], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Sum[(-3)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!
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PROG
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(Magma) [1] cat [(&+[((-3)^k*StirlingSecond(m, k)):k in [0..m]]):m in [1..25]]; // Marius A. Burtea, Jul 27 2019
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A309085
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a(n) = exp(4) * Sum_{k>=0} (-4)^k*k^n/k!.
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4
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1, -4, 12, -20, -20, 172, 108, -2388, -3220, 47532, 161900, -1062740, -8532628, 13623212, 431041132, 1206169260, -17833021588, -169685043796, 180187176044, 13462762665132, 79377664422252, -553096696140884, -11670986989785492, -44371854928405844, 829755609457185644
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Sum_{j>=0} (-4)^j*x^j / Product_{k=1..j} (1 - k*x).
E.g.f.: exp(4*(1 - exp(x))).
a(n) = Sum_{k=0..n} (-4)^k * Stirling2(n,k).
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MATHEMATICA
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Table[Exp[4] Sum[(-4)^k k^n/k!, {k, 0, Infinity}], {n, 0, 24}]
Table[BellB[n, -4], {n, 0, 24}]
nmax = 24; CoefficientList[Series[Sum[(-4)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[4 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!
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PROG
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(Magma) [1] cat [(&+[((-4)^k*StirlingSecond(m, k)):k in [0..m]]):m in [1..24]]; // Marius A. Burtea, Jul 11 2019
(PARI) a(n) = sum(k=0, n, (-4)^k * stirling(n, k, 2)); \\ Michel Marcus, Jul 12 2019
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A309386
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Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} (-k)^(n-j)*Stirling2(n,j).
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1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 1, 1, 1, 1, -3, 1, 9, 2, 1, 1, 1, -4, 5, 19, -23, -9, 1, 1, 1, -5, 11, 25, -128, -25, 9, 1, 1, 1, -6, 19, 21, -343, 379, 583, 50, 1, 1, 1, -7, 29, 1, -674, 2133, 1549, -3087, -267, 1
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OFFSET
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0,18
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LINKS
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FORMULA
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E.g.f. of column k: exp((1 - exp(-k*x))/k) for k > 0.
A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} (-k)^(n-1-j) * binomial(n-1,j) * A(j,k) for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 0, -1, -2, -3, -4, -5, ...
1, -1, -1, 1, 5, 11, 19, ...
1, 1, 9, 19, 25, 21, 1, ...
1, 2, -23, -128, -343, -674, -1103, ...
1, -9, -25, 379, 2133, 6551, 15211, ...
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MATHEMATICA
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T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * StirlingS2[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 07 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A335977
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Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(1 - exp(x)) + x).
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7
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1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -1, -1, 1, 1, -3, 1, 3, 2, 1, 1, -4, 5, 7, 7, 9, 1, 1, -5, 11, 5, -8, -13, 9, 1, 1, -6, 19, -9, -43, -65, -89, -50, 1, 1, -7, 29, -41, -74, -27, 37, -45, -267, 1, 1, -8, 41, -97, -53, 221, 597, 1024, 1191, -413, 1, 1, -9, 55, -183, 92, 679, 961, 805, 1351, 4723, 2180, 1
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OFFSET
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0,12
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LINKS
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FORMULA
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T(0,k) = 1 and T(n,k) = T(n-1,k) - k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
T(n,k) = exp(k) * Sum_{j>=0} (j + 1)^n * (-k)^j / j!.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 0, -1, -2, -3, -4, -5, ...
1, -1, -1, 1, 5, 11, 19, ...
1, -1, 3, 7, 5, -9, -41, ...
1, 2, 7, -8, -43, -74, -53, ...
1, 9, -13, -65, -27, 221, 679, ...
1, 9, -89, 37, 597, 961, -341, ...
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MATHEMATICA
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T[0, k_] := 1; T[n_, k_] := T[n - 1, k] - k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A350263
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Triangle read by rows. T(n, k) = BellPolynomial(n, -k).
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9
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1, 0, -1, 0, 0, 2, 0, 1, 2, -3, 0, 1, -6, -21, -20, 0, -2, -14, 24, 172, 370, 0, -9, 26, 195, 108, -1105, -4074, 0, -9, 178, -111, -2388, -4805, 2046, 34293, 0, 50, 90, -3072, -3220, 23670, 87510, 111860, -138312, 0, 267, -2382, -4053, 47532, 121995, -115458, -1193157, -2966088, -2932533
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OFFSET
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0,6
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LINKS
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EXAMPLE
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[0] 1
[1] 0, -1
[2] 0, 0, 2
[3] 0, 1, 2, -3
[4] 0, 1, -6, -21, -20
[5] 0, -2, -14, 24, 172, 370
[6] 0, -9, 26, 195, 108, -1105, - 4074
[7] 0, -9, 178, -111, -2388, -4805, 2046, 34293
[8] 0, 50, 90, -3072, -3220, 23670, 87510, 111860, -138312
[9] 0, 267, -2382, -4053, 47532, 121995, -115458, -1193157, -2966088, -2932533
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MAPLE
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A350263 := (n, k) -> ifelse(n = 0, 1, BellB(n, -k)):
seq(seq(A350263(n, k), k = 0..n), n = 0..9);
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MATHEMATICA
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T[n_, k_] := BellB[n, -k]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A351776
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.
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1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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E.g.f. of column k: 1/(1 + k*x*exp(x)).
T(0,k) = 1 and T(n,k) = -k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, -1, -2, -3, -4, -5, ...
0, 0, 4, 12, 24, 40, ...
0, 3, -6, -63, -204, -465, ...
0, -4, -8, 420, 2288, 7180, ...
0, -25, 150, -3435, -32020, -138525, ...
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PROG
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(PARI) T(n, k) = n!*sum(j=0, n, (-k)^(n-j)*(n-j)^j/j!);
(PARI) T(n, k) = if(n==0, 1, -k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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