A063170 -id:A063170 - cp's OEIS frontend

A354943 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * n^(n-k).

1, 2, 22, 438, 12824, 496370, 23914512, 1379269094, 92667551104, 7108231236066, 612974464428800, 58702772664490262, 6181602019316333568, 709911177607125141362, 88301595129435811723264, 11825985945777638231211750, 1696696168760520436580974592, 259624546758869333450285984066

Offset: 0

Ilya Gutkovskiy, Jun 12 2022

nonn

Cf. A000172, A063170, A216831, A241247, A274246, A277373, A277386, A354944.

Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k]^3 k! n^(n - k), {k, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[n!^3 SeriesCoefficient[BesselI[0, 2 Sqrt[x]] Sum[n^k x^k/k!^3, {k, 0, n}], {x, 0, n}], {n, 0, 17}]

a(n) = sum(k=0, n, binomial(n,k)^3 * k! * n^(n-k)); \\ Michel Marcus, Jun 12 2022

a(n) = n!^3 * [x^n] BesselI(0,2*sqrt(x)) * Sum_{k>=0} n^k * x^k / k!^3.

a(n) ~ c * n^(n - 1/2) / (exp(r*n) * r^(2*n)), where r = (2 - 5*(2/(3*sqrt(69)-11))^(1/3) + ((3*sqrt(69)-11)/2)^(1/3))/3 = 0.430159709001946734... is the real root of the equation r^2 = (1-r)^3 and c = sqrt(138 + 2^(2/3)*(69*(8901 - 223*sqrt(69)))^(1/3) + 2^(2/3)*(69*(8901 + 223*sqrt(69)))^(1/3))/(2*sqrt(69*Pi)) = 0.684738330749970434111338151096549475398274404060139170789278633219363118... - Vaclav Kotesovec, Jul 01 2022, updated Mar 17 2024

A134558 Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 24, 16, 10, 4, 1, 120, 65, 38, 17, 5, 1, 720, 326, 168, 78, 26, 6, 1, 5040, 1957, 872, 393, 142, 37, 7, 1, 40320, 13700, 5296, 2208, 824, 236, 50, 8, 1, 362880, 109601, 37200, 13977, 5144, 1569, 366, 65, 9, 1, 3628800, 986410, 297856

Offset: 0

Ross La Haye, Jan 22 2008

			Square array begins:
    1,    1,    1,     1,     1,     1,      1, ...
    1,    2,    3,     4,     5,     6,      7, ...
    2,    5,   10,    17,    26,    37,     50, ...
    6,   16,   38,    78,   142,   236,    366, ...
   24,   65,  168,   393,   824,  1569,   2760, ...
  120,  326,  872,  2208,  5144, 10970,  21576, ...
  720, 1957, 5296, 13977, 34960, 81445, 176112, ...

Eric Weisstein's World of Mathematics, Incomplete Gamma Function.
Wikipedia, Incomplete gamma function.

Cf. a(n, 0) = A000142(n); a(n, 1) = A000522(n); a(n, 2) = A010842(n); a(n, 3) = A053486(n); a(n, 4) = A053487(n); a(n, 5) = A080954(n); a(n, 6) = A108869(n); a(1, k) = A000027(k+1); a(2, k) = A002522(k+1); a(n, n) = A063170(n); a(n, n+1) = A001865(n+1); a(n, n+2) = A001863(n+2).
Another version: A089258.
A transposed version: A080955.
Cf. A001113.

T[n_,k_] := Gamma[n+1, k]*E^k; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] //Flatten (* Amiram Eldar, Jun 27 2020 *)

a(n,k) = gamma(n+1,k)*e^k = Sum_{m=0..n} m!*binomial(n,m)*k^(n-m).
a(n,k) = n*a(n-1,k) + k^n for n,k > 0.
E.g.f. (by columns) is e^(kx)/(1-x).
a(n,k) = the binomial transform by columns of a(n,k-1).
Conjecture: a(n,k) is the permanent of the n X n matrix with k+1 on the main diagonal and 1 elsewhere.

More terms from Amiram Eldar, Jun 27 2020

A226931 Numerator of n + Sum(binomial(n,k)(k/n)^k((n-k)/n)^(n-k), k=0..n).

Original entry on oeis.org

3, 9, 53, 231, 5319, 3167, 1296273, 1604979, 64370707, 22906587, 411169704813, 610433321, 424312831956207, 2146177886409, 98731231639051, 12218411169233691, 1112291237880234922707, 2196818399875253, 2619031544578888560315813, 16827894135040576041

Offset: 1

N. J. A. Sloane, Jul 31 2013

			3, 9/2, 53/9, 231/32, 5319/625, 3167/324, 1296273/117649, 1604979/131072, ...

Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7. See xi_2(n).

Denominators are in A036505. Cf. A090878, A063170.

a(n) = numerator(n + sum(k=0, n, binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k))); \\ Michel Marcus, Jun 11 2015

A336999 a(n) = n! * Sum_{d|n} n^d / d!.

Original entry on oeis.org

1, 8, 45, 544, 3725, 89856, 858823, 25271296, 434776329, 13241728000, 285750755411, 11494661861376, 302956057862653, 12945137688641536, 446924199188379375, 20735627677666902016, 827246308572614396177, 43155924331583693389824

Offset: 1

Ilya Gutkovskiy, Aug 10 2020

nonn

Cf. A057625, A063170, A066108.

Table[n! Sum[n^d/d!, {d, Divisors[n]}], {n, 1, 18}]
Table[n! SeriesCoefficient[Sum[(Exp[n x^k] - 1), {k, 1, n}], {x, 0, n}], {n, 1, 18}]

a(n) = n! * sumdiv(n, d, n^d/d!); \\ Michel Marcus, Aug 12 2020

a(n) = n! * [x^n] Sum_{k>=1} (exp(n*x^k) - 1).

cp's OEIS Frontend

A354943 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * n^(n-k).

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A134558 Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...

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A226931 Numerator of n + Sum(binomial(n,k)(k/n)^k((n-k)/n)^(n-k), k=0..n).

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PARI

A336999 a(n) = n! * Sum_{d|n} n^d / d!.

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cp's OEIS Frontend

A354943 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * n^(n-k).

Views

Author

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Crossrefs

Programs

Mathematica

PARI

Formula

A134558 Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...

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Author

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Examples

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Programs

Mathematica

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Extensions

A226931 Numerator of n + Sum(binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k), k=0..n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A336999 a(n) = n! * Sum_{d|n} n^d / d!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A226931 Numerator of n + Sum(binomial(n,k)(k/n)^k((n-k)/n)^(n-k), k=0..n).