A039814 -id:A039814 - cp's OEIS frontend

A341588 E.g.f.: -log(1 + log(1 - x))^3 / 6.

1, 12, 130, 1485, 18508, 253400, 3805723, 62437500, 1113510409, 21479997957, 446094038806, 9930796412082, 236037249893092, 5968192832899412, 160007282538148508, 4534905316824903144, 135500246340709682692, 4257646241716404353684, 140366073694357927723936, 4845119946789226304526392

Offset: 3

Ilya Gutkovskiy, Feb 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 3..418

Cf. A000399, A000559, A003713, A008275, A039814, A341587.

nmax = 22; CoefficientList[Series[-Log[1 + Log[1 - x]]^3/6, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
Table[Sum[Abs[StirlingS1[n, k] StirlingS1[k, 3]], {k, 3, n}], {n, 3, 22}]

a(n) = Sum_{k=3..n} |Stirling1(n, k) * Stirling1(k, 3)|.

a(n) ~ (n-1)! * log(n)^2 / (2 * (1 - exp(-1))^n) * (1 + (2*gamma - 2*log(exp(1) - 1)) / log(n) + (gamma^2 - Pi^2/6 - 2*log(exp(1) - 1)*gamma + log(exp(1)-1)^2) / log(n)^2). - Vaclav Kotesovec, Jun 04 2022

A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 19, 8, 1, 14, 55, 65, 16, 1, 20, 125, 285, 211, 32, 1, 27, 245, 910, 1351, 665, 64, 1, 35, 434, 2380, 5901, 6069, 2059, 128, 1, 44, 714, 5418, 20181, 35574, 26335, 6305, 256, 1, 54, 1110, 11130, 58107, 156660, 204205, 111645, 19171, 512

Offset: 1

Thomas Wieder, Dec 29 2006

Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row A000079 = the powers of two = 2^n. As the second row we have A001047 = 3^n - 2^n. As the column sums we have 1,3,10,37,151,674,3263,17007,94828 we have A005493 = number of partitions of [n+1] with a distinguished block.

			Matrix begins:
1, 2, 4,  8, 16,  32,   64,  128,   256, ... A000079
0, 1, 5, 19, 65, 211,  665, 2059,  6305, ... A001047
0, 0, 1,  9, 55, 285, 1351, 6069, 26335, ... A016269
0, 0, 0,  1, 14, 125,  910, 5901, 35574, ... A025211
0, 0, 0,  0,  1,  20,  245, 2380, 20181, ...
0, 0, 0,  0,  0,   1,   27,  434,  5418, ...
0, 0, 0,  0,  0,   0,    1,   35,   714, ...
0, 0, 0,  0,  0,   0,    0,    1,    44, ...
0, 0, 0,  0,  0,   0,    0,    0,     1, ...
Triangle begins:
1;
1,  2;
1,  5,  4;
1,  9, 19,  8;
1, 14, 55, 65, 16;

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A039810, A039814, A126350, A054654, A126353.

T:= (n, k)-> add(binomial(n-1, i-1) *Stirling2(i, n+1-k), i=1..n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 29 2011

T[n_, k_] := Sum[Binomial[n-1, i-1]*StirlingS2[i, n+1-k], {i, 1, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

(In Maple notation:) Matrix product B.A of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling2(j,i) with i from 1 to d, j from 1 to d, d=9.

T(n,k) = Sum_{i=1..n} C(n-1,i-1) * Stirling2(i, n+1-k). - Alois P. Heinz, Sep 29 2011

A126350 Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the second kind.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 1, 9, 22, 15, 1, 14, 61, 99, 52, 1, 20, 135, 385, 471, 203, 1, 27, 260, 1140, 2416, 2386, 877, 1, 35, 455, 2835, 9156, 15470, 12867, 4140, 1, 44, 742, 6230, 28441, 72590, 102215, 73681, 21147

Offset: 1

Thomas Wieder, Dec 29 2006

Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row (not surprisingly) A000110 = Bell or exponential numbers: ways of placing n labeled balls into n indistinguishable boxes . As second row we have A033452 = "STIRLING" transform of squares A000290. As the column sums we have 1, 3, 11, 47, 227, 1215, 7107, 44959, 305091 which is A035009 = STIRLING transform of [1,1,2,4,8,16,32, ...].

			Matrix begins:
1 2 5 15 52 203  877  4140  21147
0 1 5 22 99 471 2386 12867  73681
0 0 1  9 61 385 2416 15470 102215
0 0 0  1 14 135 1140  9156  72590
0 0 0  0 1   20  260  2835  28441
0 0 0  0 0    1   27   455   6230
0 0 0  0 0    0    1    35    742
0 0 0  0 0    0    0     1     44
0 0 0  0 0    0    0     0      1

Cf. A039810, A039814, A126351, A054654, A126353.

Cf. A137597.

T:= (n, k)-> add(Stirling2(n, j)*binomial(j-1, n-k), j=n-k+1..n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 03 2019

T[dim_] := T[dim] = Module[{M}, M[n_, n_] = 1; M[, ] = 0; Do[M[n, k] = M[n-1, k-1] + (k+2) M[n-1, k] + (k+1) M[n-1, k+1], {n, 0, dim-1}, {k, 0, n-1}]; Array[M, {dim, dim}, {0, 0}]];
dim = 9;
Table[T[dim][[n]][[1 ;; n]] // Reverse, {n, 1, dim}] (* Jean-François Alcover, Jun 27 2019, from Sage *)

def A126350_triangle(dim): # rows in reversed order
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(k+2)*M[n-1,k]+(k+1)*M[n-1,k+1]
    return M
A126350_triangle(9) # Peter Luschny, Sep 19 2012

(In Maple notation:) Matrix product A.B of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling2(j,i) with i from 1 to d, j from 1 to d, d=9.

A126353 Triangle read by rows: matrix product of the Stirling numbers of the first kind with the binomial coefficients.

Original entry on oeis.org

1, 1, 0, 1, -1, 1, 1, -3, 5, -2, 1, -6, 17, -20, 9, 1, -10, 45, -100, 109, -44, 1, -15, 100, -355, 694, -689, 265, 1, -21, 196, -1015, 3094, -5453, 5053, -1854, 1, -28, 350, -2492, 10899, -29596, 48082, -42048, 14833

Offset: 1

Thomas Wieder, Dec 29 2006

Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row A000166 = subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

			Matrix begins:
1 0 1 -2 9 -44 265 -1854 14833
0 1 -1 5 -20 109 -689 5053 -42048
0 0 1 -3 17 -100 694 -5453 48082
0 0 0 1 -6 45 -355 3094 -29596
0 0 0 0 1 -10 100 -1015 10899
0 0 0 0 0 1 -15 196 -2492
0 0 0 0 0 0 1 -21 350
0 0 0 0 0 0 0 1 -28
0 0 0 0 0 0 0 0 1

Signed version of A094791 [from Olivier Gérard, Jul 31 2011]

Cf. A039810, A039814, A126350, A126351, A054654.

(In Maple notation:) Matrix product B.A of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.

A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 7, -6, 1, 0, -35, 40, -12, 1, 0, 228, -315, 130, -20, 1, 0, -1834, 2908, -1485, 320, -30, 1, 0, 17582, -30989, 18508, -5005, 665, -42, 1, 0, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1

Offset: 0

Peter Luschny, Jun 27 2019

			Triangle starts:
[0] [1]
[1] [0,       1]
[2] [0,      -2,        1]
[3] [0,       7,       -6,       1]
[4] [0,     -35,       40,     -12,        1]
[5] [0,     228,     -315,     130,      -20,      1]
[6] [0,   -1834,     2908,   -1485,      320,    -30,      1]
[7] [0,   17582,   -30989,   18508,    -5005,    665,    -42,    1]
[8] [0, -195866,   375611, -253400,    81088, -13650,   1232,  -56,   1]
[9] [0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1]

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, A new set of Sheffer-Bell polynomials and logarithmic numbers, Georgian Mathematical Journal, Feb. 2019, page 8.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.

Columns k=0..3 give A000007, (-1)^(n+1) * A003713(n), (-1)^n * A341587(n), (-1)^(n+1) * A341588(n).

Cf. A039814 (variant), A129062, A325873.

p[n_] := Sum[StirlingS1[n, k] FactorialPower[x, k] , {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten

T(n, k) = sum(j=k, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 18 2025

def a_row(n):
    s = sum((-1)^(n-k)*stirling_number1(n,k)*falling_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)]

From Seiichi Manyama, Apr 18 2025: (Start)
T(n,k) = Sum_{j=k..n} Stirling1(n,j) * Stirling1(j,k).
E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 + log(1 + x)). (End)

A341589 a(n) = Sum_{k=n..2n} |Stirling1(2n, k) * Stirling1(k, n)|.

Original entry on oeis.org

1, 2, 40, 1485, 81088, 5856900, 526685269, 56704848200, 7112345477952, 1018548226480356, 163987811350464660, 29321558852248050388, 5764958268855541178967, 1236150756215397667568170, 287086392921014590422630300, 71789589754855255636302048525, 19231403740347427723119910379040

Offset: 0

Ilya Gutkovskiy, Feb 15 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A003713, A008275, A039814, A321712, A341587, A341588.

Table[Sum[Abs[StirlingS1[2 n, k] StirlingS1[k, n]], {k, n, 2 n}], {n, 0, 16}]
Table[((2 n)!/n!) SeriesCoefficient[(-Log[1 + Log[1 - x]])^n, {x, 0, 2 n}], {n, 0, 16}]

a(n) = sum(k=n, 2*n, abs(stirling(2*n, k, 1)*stirling(k, n, 1))); \\ Michel Marcus, Feb 16 2021

a(n) = ((2*n)!/n!) * [x^(2*n)] (-log(1 + log(1 - x)))^n.
From Vaclav Kotesovec, Feb 15 2021: (Start)
a(n) ~ c * d^n * (n-1)!, where
d = -16*p*q^2 * log(-2*q/(1+r))^(1+r) / ((1 + 2*q + r)^2 * (1 + 1/(p*(1+r)))^r) = 17.84101281316291323354184111891200669611476053165484517795417711039479218...
p = LambertW(-1, -1/(exp(1/(1+r))*(1+r)))
q = LambertW(-1, -(1+r)/exp((1+r)/2)/2)
r = 0.5094050884976689299791685259225203723646676600942448390861428232759777841...
is the root of the equation (1+p)*(1+r)^2 * (1 + 2*q + r) * log(-p*(1+r)) + 2*log(-(1+r)/(2*q)) * ((1+q)*(1 + p + p*r) - (1+r) * log(-p*(1+r)) * (p - q + r + p*r + (1+p) * (1+q) * (1+r) * (log(1 + 1/(p*(1+r))) - log(-log(-(1+r)/(2*q)))))) = 0
and c = 0.1417076025518808268972093339771762801784527709... (End)

A142472 Triangle T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k), read by rows.

Original entry on oeis.org

1, -4, 1, 21, -18, 1, -140, 240, -48, 1, 1140, -3150, 1300, -100, 1, -11004, 43620, -29700, 4800, -180, 1, 123074, -650769, 647780, -175175, 13965, -294, 1, -1566928, 10517108, -14190400, 5676160, -764400, 34496, -448, 1, 22390488, -184052520, 319680732, -175091112, 35160048, -2698920, 75600, -648, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 22 2008

Row sums are: 1, -3, 4, 53, -809, 7537, -41418, -294411, 15463669, -352665269, ....

			The triangle begins as:
         1;
        -4,          1;
        21,        -18,         1;
      -140,        240,       -48,          1;
      1140,      -3150,      1300,       -100,        1;
    -11004,      43620,    -29700,       4800,     -180,        1;
    123074,    -650769,    647780,    -175175,    13965,     -294,     1;
  -1566928,   10517108, -14190400,    5676160,  -764400,    34496,  -448,    1;
  22390488, -184052520, 319680732, -175091112, 35160048, -2698920, 75600, -648, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A008275, A039814.

A142472:= func< n,k | Binomial(n,k)*(&+[StirlingFirst(n,j)*StirlingFirst(j,k): j in [k..n]]) >;
[A142472(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 02 2021

A142472:= (n,k)-> binomial(n,k)*add(Stirling1(n,j)*Stirling1(j,k), j=k..n);
seq(seq(A142472(n,k), k=1..n), n=1..12); # G. C. Greubel, Apr 02 2021

T[n_, k_]:= Binomial[n, k]*Sum[StirlingS1[n, j]*StirlingS1[j, k], {j, k, n}];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Apr 02 2021 *)

def A142472(n,k): return (-1)^(n-k)*binomial(n,k)*sum( stirling_number1(n, j)*stirling_number1(j, k) for j in (k..n) )
flatten([[A142472(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 02 2021

T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k).

Edited by N. J. A. Sloane, Sep 26 2008

cp's OEIS Frontend

A341588 E.g.f.: -log(1 + log(1 - x))^3 / 6.

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A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

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A126350 Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the second kind.

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A126353 Triangle read by rows: matrix product of the Stirling numbers of the first kind with the binomial coefficients.

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A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

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A341589 a(n) = Sum_{k=n..2*n} |Stirling1(2*n, k) * Stirling1(k, n)|.

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A142472 Triangle T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k), read by rows.

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A341589 a(n) = Sum_{k=n..2n} |Stirling1(2n, k) * Stirling1(k, n)|.