| Displaying 1-9 of 9 results found.
page
1
a(n) = B(n,n), where B(n,x) = Sum_{k=0..n} Stirling2(n,k)*x^k are the Bell polynomials (also known as exponential polynomials or Touchard polynomials).
+10
23
1, 1, 6, 57, 756, 12880, 268098, 6593839, 187104200, 6016681467, 216229931110, 8588688990640, 373625770888956, 17666550789597073, 902162954264563306, 49482106424507339565, 2901159958960121863952, 181069240855214001514460, 11985869691525854175222222
FORMULA
E.g.f.: x*f'(x)/f(x), where f(x) is the generating series for sequence A035051. a(n) ~ (exp(1/LambertW(1)-2)/LambertW(1))^n * n^n / sqrt(1+LambertW(1)). - Vaclav Kotesovec, May 23 2014 Conjecture: It appears that the equation a(x)*e^x = Sum_{n=0..oo} ( (n^x*x^n)/n! ) is true for every positive integer x. - Nicolas Nagel, Apr 20 2016 [This is just the special case k=x of the formula B(k,x) = e^(-x) * Sum_{n=0..oo} n^k*x^n/n!; see for example the World of Mathematics link. - Pontus von Brömssen, Dec 05 2020] a(n) = [x^n] Sum_{k=0..n} n^k*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, May 31 2018
MAPLE
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:
a:= n-> A(n$2):
MATHEMATICA
Table[BellB[n, n], {n, 0, 100}]
PROG
(Maxima) a(n):=stirling2(n, 0)+sum(stirling2(n, k)*n^k, k, 1, n);
makelist(a(n), n, 0, 30);
(PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*n^k); \\ Michel Marcus, Apr 20 2016
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*(1 - exp(x))).
+10
9
1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, 1, 0, 1, -4, 6, 2, 1, 0, 1, -5, 12, -3, -6, -2, 0, 1, -6, 20, -20, -21, -14, -9, 0, 1, -7, 30, -55, -20, 24, 26, -9, 0, 1, -8, 42, -114, 45, 172, 195, 178, 50, 0, 1, -9, 56, -203, 246, 370, 108, -111, 90, 267, 0, 1, -10, 72, -328, 679, 318, -1105, -2388, -3072, -2382, 413, 0
FORMULA
A(0,k) = 1 and A(n,k) = -k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} (-k)^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, -1, -2, -3, -4, -5, -6, ...
0, 0, 2, 6, 12, 20, 30, ...
0, 1, 2, -3, -20, -55, -114, ...
0, 1, -6, -21, -20, 45, 246, ...
0, -2, -14, 24, 172, 370, 318, ...
0, -9, 26, 195, 108, -1105, -4074, ...
MAPLE
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:
MATHEMATICA
A[n_, k_] := Sum[(-k)^j StirlingS2[n, j], {j, 0, n}];
A292861[n_, k_] := BellB[k, k - n];
Triangle read by rows. T(n, k) = BellPolynomial(n, -k).
+10
9
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
EXAMPLE
[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
MAPLE
A350263 := (n, k) -> ifelse(n = 0, 1, BellB(n, -k)):
seq(seq( A350263(n, k), k = 0..n), n = 0..9);
MATHEMATICA
T[n_, k_] := BellB[n, -k]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + n*j*x).
+10
8
1, 1, -1, 1, 25, -674, 15211, -331827, 5987745, 15901597, -13125035449, 1292056076070, -103145930581319, 7462324963409941, -464957409070517453, 16313974895147212801, 2059903411953959582849, -708700955022151333496910, 143215213612865558214820303, -24681846509158429152517973103
FORMULA
a(n) = n! * [x^n] exp((1 - exp(-n*x))/n), for n > 0.
a(n) = Sum_{k=0..n} (-n)^(n-k)*Stirling2(n,k).
a(n) = (-n)^n*BellPolynomial_n(-1/n) for n >= 1. - Peter Luschny, Aug 20 2018
MATHEMATICA
Table[SeriesCoefficient[Sum[x^k/Product[(1 + n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(1 - Exp[-n x])/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[(-n)^(n - k) StirlingS2[n, k], {k, n}], {n, 19}]]
Join[{1}, Table[(-n)^n BellB[n, -1/n], {n, 1, 21}]] (* Peter Luschny, Aug 20 2018 *)
PROG
(PARI) {a(n) = sum(k=0, n, (-n)^(n-k)*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 27 2019
a(n) = exp(n) * Sum_{k>=0} (k + 1)^n * (-n)^k / k!.
+10
7
1, 0, -1, 7, -43, 221, -341, -15980, 370761, -5688125, 62689871, -197586839, -14973562979, 585250669316, -14306382821485, 240985102271971, -1121421968408303, -122020498882279931, 6674724196051810807, -223424819176020519168, 5051515662105879438501
FORMULA
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (-n*x/(1 - x))^k / Product_{j=1..k} (1 - j*x/(1 - x)).
a(n) = n! * [x^n] exp(x + n*(1 - exp(x))).
a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(-n).
MATHEMATICA
Table[n! SeriesCoefficient[Exp[x + n (1 - Exp[x])], {x, 0, n}], {n, 0, 20}]
Table[Sum[Binomial[n, k] BellB[k, -n], {k, 0, n}], {n, 0, 20}]
a(n) = exp(n) * Sum_{k>=0} (-n)^k * (k - 1)^n / k!.
+10
5
1, -2, 7, -31, 149, -631, 475, 43210, -844727, 10960505, -86569889, -584746911, 46302579229, -1304510879686, 25366896568707, -277053418780891, -4271166460501743, 384590020131637825, -14617527176248527545, 380117694164438489422, -5265650620303861935579
FORMULA
a(n) = n! * [x^n] exp(n*(1 - exp(x)) - x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * BellPolynomial_k(-n).
MATHEMATICA
Table[n! SeriesCoefficient[Exp[n (1 - Exp[x]) - x], {x, 0, n}], {n, 0, 20}]
Table[Sum[(-1)^(n - k) Binomial[n, k] BellB[k, -n], {k, 0, n}], {n, 0, 20}]
a(n) = exp(n) * Sum_{k>=0} (k + n)^n * (-n)^k / k!.
+10
4
1, 0, -2, -3, 44, 245, -2346, -33278, 186808, 6888555, -6774910, -1986368439, -10227075420, 738830661296, 10363304656782, -327255834908715, -9380517430358288, 152180429032236325, 9132761207739810618, -46897839494116200918, -9833058047657527541220
FORMULA
a(n) = n! * [x^n] exp(n*(1 + x - exp(x))).
a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(-n) * n^(n-k).
MATHEMATICA
Table[n! SeriesCoefficient[Exp[n (1 + x - Exp[x])], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[k, -n] n^(n - k), {k, 0, n}], {n, 1, 20}]]
a(n) = exp(n) * Sum_{k>=0} (-1)^k * n^(k-1) * k^(n-1) / k!.
+10
2
1, -1, 2, -5, 9, 53, -1107, 12983, -116470, 560049, 8370713, -346902877, 7551856337, -117404648467, 913399734614, 22560135521007, -1393700803877939, 44331044030953865, -979905458659247779, 10462396536804802459, 367799071887303276422, -30046998012662824941947
FORMULA
a(n) = Sum_{k=0..n-1} (-1)^k * Stirling2(n-1,k) * n^(k-1).
a(n) = BellPolynomial_(n-1)(-n) / n.
MATHEMATICA
Table[Sum[(-1)^k StirlingS2[n - 1, k] n^(k - 1), {k, 0, n - 1}], {n, 1, 22}]
Table[BellB[n - 1, -n]/n, {n, 1, 22}]
PROG
(PARI) a(n)={sum(k=0, n-1, (-1)^k * stirling(n-1, k, 2) * n^(k-1))} \\ Andrew Howroyd, May 18 2020
a(n) = (-1)^n * exp(n) * Sum_{k>=1} (-1)^k * n^(k-1) * k^n / k!.
+10
0
1, 1, 1, -5, -74, -679, -4899, -17289, 325837, 10627109, 199348590, 2684041427, 15872610469, -546948563407, -27499774835519, -778467357484561, -15311413773551790, -125363405319188419, 6452292137017871097, 436442148982835915339, 16494863323310244977581
FORMULA
E.g.f.: series reversion of -log(1 - x) * exp(-x).
a(n) = (n - 1)! * [x^n] exp(n*(1 - exp(-x))).
a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling2(n,k) * n^(k-1).
a(n) = (-1)^n * BellPolynomial_n(-n) / n.
MATHEMATICA
nmax = 21; CoefficientList[InverseSeries[Series[-Log[1 - x] Exp[-x], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest
Table[Sum[(-1)^(n - k) StirlingS2[n, k] n^(k - 1), {k, 1, n}], {n, 1, 21}]
Table[(-1)^n BellB[n, -n]/n, {n, 1, 21}]
PROG
(PARI) a(n) = sum(k=1, n, (-1)^(n-k) * stirling(n, k, 2) * n^(k-1)); \\ Michel Marcus, Apr 20 2020
Search completed in 0.011 seconds
| 