A061711 -id:A061711 - cp's OEIS frontend

A331988 Table T(n,k) read by antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}.

1, 2, 2, 6, 9, 3, 24, 64, 20, 4, 120, 625, 216, 36, 5, 720, 7776, 3136, 512, 56, 6, 5040, 117649, 59049, 10000, 1000, 81, 7, 40320, 2097152, 1331000, 248832, 24336, 1728, 110, 8, 362880, 43046721, 35831808, 7529536, 759375, 50625, 2744, 144, 9, 3628800, 1000000000, 1097199376, 268435456, 28652616, 1889568, 93636, 4096, 182, 10

Offset: 1

Chai Wah Wu, Feb 23 2020

A dual sequence to A260355. See arXiv link for sets of permutations that achieve the value of T(n,k). The minimum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] is equal to n!*k^n.

			T(n,k)
   k    1    2     3      4      5      6      7      8      9     10     11     12
  ---------------------------------------------------------------------------------
n  1|   1    2     3      4      5      6      7      8      9     10     11     12
   2|   2    9    20     36     56     81    110    144    182    225    272    324
   3|   6   64   216    512   1000   1728   2744   4096   5832   8000  10648  13824
   4|  24  625  3136  10000  24336  50625  93636 160000 256036 390625 571536 810000

Chai Wah Wu, Table of n, a(n) for n = 1..70
Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized, arXiv:1508.02934 [math.CO], 2015-2020.
Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Cf. A000027, A000142, A000169, A001504, A016743, A016766, A061711, A061718, A260355.

from itertools import permutations, combinations_with_replacement
def A331988(n,k): # compute T(n,k)
    if k == 1:
        count = 1
        for i in range(1,n):
            count *= i+1
        return count
    ntuple, count = tuple(range(1,n+1)), 0
    for s in combinations_with_replacement(permutations(ntuple,n),k-2):
        t = list(ntuple)
        for d in s:
            for i in range(n):
                t[i] += d[i]
        t.sort()
        w = 1
        for i in range(n):
            w *= (n-i)+t[i]
        if w > count:
            count = w
    return count

T(n,n) = (n*(n+1)/2)^n = A061718(n).
T(n,k) <= (k(n+1)/2)^n.
T(1,k) = k = A000027(k).
T(n,1) = n! = A000142(n).
T(2,2m) = 9m^2 = A016766(m).
T(2,2m+1) = (3m+1)*(3m+2) = A001504(m).
T(n,2) = (n+1)^n = A000169(n+1).
T(3,k) = 8k^3 = A016743(k) for k > 1.
If n divides k then T(n,k) = (k*(n+1)/2)^n.
If k is even then T(n,k) = (k*(n+1)/2)^n.
If n is odd and k >= n-1 then T(n,k) = (k*(n+1)/2)^n.
If n is even and k is odd such that k >= n-1, then T(n,k) = ((k^2*(n+1)^2-1)/4)^(n/2).

A350149 Triangle read by rows: T(n, k) = n^(n-k)*k!.

Original entry on oeis.org

1, 1, 1, 4, 2, 2, 27, 9, 6, 6, 256, 64, 32, 24, 24, 3125, 625, 250, 150, 120, 120, 46656, 7776, 2592, 1296, 864, 720, 720, 823543, 117649, 33614, 14406, 8232, 5880, 5040, 5040, 16777216, 2097152, 524288, 196608, 98304, 61440, 46080, 40320, 40320

Offset: 0

Robert B Fowler, Dec 27 2021

T(n,k) are the denominators in a double summation power series for the definite integral of x^x. First expand x^x = exp(x*log(x)) = Sum_{n>=0} (x*log(x))^n/n!, then integrate each of the terms to get the double summation for F(x) = Integral_{t=0..x} t^t = Sum_{n>=1} (Sum_{k=0..n-1} (-1)^(n+k+1)*x^n*(log(x))^k/T(n,k)).
This is a definite integral, because lim {x->0} F(x) = 0.
The value of F(1) = 0.78343... = A083648 is known humorously as the Sophomore's Dream (see Borwein et al.).

			Triangle T(n,k) begins:
--------------------------------------------------------------------------
n/k         0        1       2       3      4      5      6      7      8
--------------------------------------------------------------------------
0  |        1,
1  |        1,       1,
2  |        4,       2,      2,
3  |       27,       9,      6,      6,
4  |      256,      64,     32,     24,    24,
5  |     3125,     625,    250,    150,   120,   120,
6  |    46656,    7776,   2592,   1296,   864,   720,   720,
7  |   823543,  117649,  33614,  14406,  8232,  5880,  5040,  5040,
8  | 16777216, 2097152, 524288, 196608, 98304, 61440, 46080, 40320, 40320.
...

Borwein, J., Bailey, D. and Girgensohn, R., Experimentation in Mathematics: Computational Paths to Discovery, A. K. Peters 2004.
William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton NJ 2005.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Sophomore's dream
Wikipedia, Sophomore's dream

Cf. A000312 (first column), A000169 (2nd column), A003308 (3rd column excluding first term), A000142 (main diagonal), A000142 (2nd diagonal excluding first term), A112541 (row sums).
Values of the integral: A083648, A073009.
Cf. A061711, A350297.

A350149:= func< n,k | n^(n-k)*Factorial(k) >;
[A350149(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 31 2022

T := (n, k) -> n^(n - k)*k!:
seq(seq(T(n, k), k = 0..n), n = 0..9); # Peter Luschny, Jan 07 2022

T[n_, k_]:= n^(n-k)*k!; Table[T[n, k], {n, 0,12}, {k,0,n}]//Flatten (* Amiram Eldar, Dec 27 2021 *)

def A350149(n,k): return n^(n-k)*factorial(k)
flatten([[A350149(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 31 2022

T(n, 0) = A000312(n).
T(n, 1) = A000169(n).
T(n, 2) = A003308(n), n >= 2.
Sum_{k=0..n} T(n, k) = A112541(n).
T(n, n) = A000142(n).
T(n, n-1) = A000142(n), n >= 1.
T(n,k) = A061711(n) * (n+1) / A350297(n+1,k). - Robert B Fowler, Jan 11 2022

A350297 Triangle read by rows: T(n,k) = n!*(n-1)^k/k!.

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 6, 12, 12, 8, 24, 72, 108, 108, 81, 120, 480, 960, 1280, 1280, 1024, 720, 3600, 9000, 15000, 18750, 18750, 15625, 5040, 30240, 90720, 181440, 272160, 326592, 326592, 279936, 40320, 282240, 987840, 2304960, 4033680, 5647152, 6588344, 6588344, 5764801

Offset: 0

Robert B Fowler, Dec 23 2021

Rows n >= 2 are coefficients in a double summation power series for the integral of x^(1/x), and the integral of its inverse function y^(y^(y^(y^(...)))). See A350358.

			Triangle T(n,k) begins:
  -----------------------------------------------------------------
   n\k     0      1      2       3       4       5       6       7
  -----------------------------------------------------------------
   0  |    1,
   1  |    1,     0,
   2  |    2,     2,     1,
   3  |    6,    12,    12,      8,
   4  |   24,    72,   108,    108,     81,
   5  |  120,   480,   960,   1280,   1280,   1024,
   6  |  720,  3600,  9000,  15000,  18750,  18750,  15625,
   7  | 5040, 30240, 90720, 181440, 272160, 326592, 326592, 279936.
  ...

Cf. A000142 (first column), A062119 (second column), A065440 (main diagonal), A055897 (subdiagonal), A217701 (row sums).
Cf. A350269, A350358.
Cf. A061711, A350149.

T := (n, k) -> (n!/k!)*(n - 1)^k:
seq(seq(T(n, k), k = 0..n), n = 0..8); # Peter Luschny, Dec 24 2021

T[1, 0] := 1; T[n_, k_] := n!*(n - 1)^k/k!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 24 2021 *)

T(n, k) = binomial(n, k)*A350269(n, k). - Peter Luschny, Dec 25 2021

T(n+1, k) = A061711(n) * (n+1) / A350149(n, k). - Robert B Fowler, Jan 11 2022

A292784 a(n) = n! * [x^n] 1/sqrt(1 - 2nx).

Original entry on oeis.org

1, 1, 12, 405, 26880, 2953125, 484989120, 111289483305, 34007836262400, 13350287284158825, 6547290750000000000, 3922838769902739011325, 2819575386162274605465600, 2394486245934541921935898125, 2371947271643716575046318080000, 2710687260280640086154937744140625, 3539907755812512418187309922385920000

Offset: 0

Ilya Gutkovskiy, Sep 23 2017

nonn

Main diagonal of A292783.

Cf. A000312, A001147, A061711, A291699.

Table[n! SeriesCoefficient[1/Sqrt[1 - 2 n x], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-i n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[n^n (2 n - 1)!!, {n, 1, 16}]]

a(n) = [x^n] 1/(1 - n*x/(1 - 2*n*x/(1 - 3*n*x/(1 - 4*n*x/(1 - 5*n*x/(1 - ...)))))), a continued fraction.

a(n) = A000312(n)*A001147(n).

A330497 a(n) = n! * Sum_{k=0..n} (-1)^k * binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 0, 1, 26, 1089, 70124, 6495985, 821315214, 136115947009, 28651724077976, 7470040450004001, 2363470644596843330, 892244303052345224641, 396227360441775922668036, 204487588996059177697597969, 121370399839482643287189048374

Offset: 0

Ilya Gutkovskiy, Dec 18 2019

nonn

Cf. A009940, A025166, A061711, A277423, A307885, A318224, A319392, A330260.

[Factorial(n)*&+[(-1)^k*Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019

Join[{1}, Table[n! Sum[(-1)^k Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, 1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[-x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]

a(n) = n! * [x^n] exp(-x/(1 - n*x)) / (1 - n*x).
a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k)^2 * n^k * k!.
a(n) ~ sqrt(2*Pi) * BesselJ(0,2) * n^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Dec 18 2019

A336765 Decimal expansion of Sum_{n>=1} 1/(n!*n^n).

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 8, 2, 9, 6, 6, 0, 0, 6, 2, 6, 3, 7, 1, 5, 0, 8, 8, 5, 2, 7, 8, 6, 9, 2, 8, 3, 4, 9, 5, 1, 3, 0, 6, 8, 1, 0, 6, 9, 5, 0, 9, 0, 2, 5, 8, 3, 9, 8, 8, 2, 1, 7, 1, 9, 5, 2, 5, 6, 0, 2, 7, 0, 9, 8, 6, 8, 8, 3, 0, 9, 2, 6, 5, 6, 8, 7, 9, 6, 0, 2, 9, 1, 8, 3, 6, 8, 6, 8, 9, 5, 5, 5, 8, 2, 5, 7, 5, 7, 7

Offset: 1

Mario Cortés, Aug 03 2020

			1.13133829660062637150885278692834951306810695090...

Cf. A061711, A073009, A094082.

evalf(sum(1/(n!*n^n), n=1..infinity), 106);  # Alois P. Heinz, Nov 20 2020

RealDigits[N[Sum[1/(n!*n^n), {n, 1, Infinity}], 800]]

suminf(n=1, 1/(n!*n^n)) \\ Michel Marcus, Aug 20 2020

Equals Sum_{n>=1} 1/A061711(n).

Missing first digit inserted by Alois P. Heinz, Nov 20 2020

A372311 Triangle read by rows: T(n, k) = n^k * Sum_{j=0..n} binomial(n - j, n - k) * Eulerian1(n, j).

Original entry on oeis.org

1, 1, 1, 1, 6, 8, 1, 21, 108, 162, 1, 60, 800, 3840, 6144, 1, 155, 4500, 48750, 225000, 375000, 1, 378, 21672, 453600, 4354560, 19595520, 33592320, 1, 889, 94668, 3500658, 60505200, 536479440, 2371803840, 4150656720

Offset: 0

Peter Luschny, Apr 26 2024

			Triangle begins:
  [0] 1;
  [1] 1,   1;
  [2] 1,   6,     8;
  [3] 1,  21,   108,     162;
  [4] 1,  60,   800,    3840,     6144;
  [5] 1, 155,  4500,   48750,   225000,    375000;
  [6] 1, 378, 21672,  453600,  4354560,  19595520,   33592320;
  [7] 1, 889, 94668, 3500658, 60505200, 536479440, 2371803840, 4150656720;

Cf. A061711 (main diagonal), A066524 (column 1), A372312 (row sums).

Cf. A163626, A173018 (eulerian1).

S := (n, k) -> local j; add(eulerian1(n, j)*binomial(n-j, n-k), j = 0..n):
row := n -> local k; seq(S(n, k) * n^k, k = 0..n):
seq(row(n), n = 0..8);

def A372311_row(n) :
    x = polygen(ZZ, 'x')
    A = []
    for m in range(0, n + 1, 1) :
        A.append((-x)^m)
        for j in range(m, 0, -1):
            A[j - 1] = j * (A[j - 1] - A[j])
    return [n^k*c for k, c in enumerate(A[0])]
for n in (0..7) : print(A372311_row(n))

A088055 a(n) = n!*n^n - ((n^(n+1)-1)/(n-1) - 1) for n>1 with a(1)=0.

Original entry on oeis.org

0, 2, 123, 5804, 371095, 33536334, 4149695921, 676438175160, 140586711200271, 36287988888888890, 11388728579602327129, 4270826370748686175140, 1886009588224061851054127, 968725766842917544760889030

Offset: 1

Amarnath Murthy, Sep 20 2003

nonn

Original definition: a(n) = G(n) - A(n), where G(n) = Sum of the first n terms of a geometric progression with first term n and common ratio n. A(n) = Product of first n terms of an arithmetic progression with first term n and common difference n.

Cf. A031972, A061711.

seq(`if`(n=1, 0, n!*n^n - ((n^(n+1)-1)/(n-1) - 1)),n=1..16); # Georg Fischer, Dec 09 2022

a(n) = if (n==1, 0, n!*n^n - ((n^(n+1)-1)/(n-1) - 1)); \\ Michel Marcus, Dec 10 2022

a(n) = A061711(n) - A031972(n) for n>1 with a(1)=0.

Corrected and extended by David Wasserman, Jun 27 2005
Edited by M. F. Hasler, Feb 12 2013
Formula negated by Georg Fischer, Dec 09 2022

A153188 Triangle read by rows: T(n,k) = n^k * k!.

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 1, 3, 18, 162, 1, 4, 32, 384, 6144, 1, 5, 50, 750, 15000, 375000, 1, 6, 72, 1296, 31104, 933120, 33592320, 1, 7, 98, 2058, 57624, 2016840, 84707280, 4150656720, 1, 8, 128, 3072, 98304, 3932160, 188743680, 10569646080, 676457349120

Offset: 0

Roger L. Bagula, Dec 20 2008

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  8;
  1, 3, 18,  162;
  1, 4, 32,  384,  6144;
  1, 5, 50,  750, 15000,  375000;
  1, 6, 72, 1296, 31104,  933120, 33592320;
  1, 7, 98, 2058, 57624, 2016840, 84707280, 4150656720;
  ...

Main diagonal gives A061711.
Row sums give A368561.
Cf. A000165, A008544.

t[n_, m_] =Product[m*k, {k, 1, n}];
Table[Table[t[n, m], {n, 1, m}], {m, 1, 10}];
Flatten[%]

T(n,k) = Product_{j=1..n} n*j.

Formula corrected by Georg Fischer, Oct 24 2024

A300519 Convolution of n! and n^n.

Original entry on oeis.org

1, 2, 7, 39, 321, 3603, 51391, 884873, 17770445, 406673247, 10431884283, 296262164637, 9224841015745, 312441152401067, 11434829066996087, 449675059390576257, 18908960744072894325, 846638474386244188311, 40213487658138717885907, 2019543479160709325145893

Offset: 0

Vaclav Kotesovec, Mar 08 2018

nonn

Cf. A000142, A000312, A061711, A277506.

Cf. A107713, A108953, A110467.

Table[Sum[If[k == 0, 1, k^k] * (n-k)!, {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} k^k * (n-k)!.

a(n) ~ n^n * (1 + exp(-1)/n).

cp's OEIS Frontend

A331988 Table T(n,k) read by antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}.

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A350149 Triangle read by rows: T(n, k) = n^(n-k)*k!.

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A350297 Triangle read by rows: T(n,k) = n!*(n-1)^k/k!.

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A292784 a(n) = n! * [x^n] 1/sqrt(1 - 2*n*x).

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A330497 a(n) = n! * Sum_{k=0..n} (-1)^k * binomial(n,k) * n^(n - k) / k!.

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A336765 Decimal expansion of Sum_{n>=1} 1/(n!*n^n).

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A372311 Triangle read by rows: T(n, k) = n^k * Sum_{j=0..n} binomial(n - j, n - k) * Eulerian1(n, j).

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A088055 a(n) = n!*n^n - ((n^(n+1)-1)/(n-1) - 1) for n>1 with a(1)=0.

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A292784 a(n) = n! * [x^n] 1/sqrt(1 - 2nx).