A103438 -id:A103438 - cp's OEIS frontend

A181134 Sum of 13th powers: a(n) = Sum_{j=0..n} j^13.

0, 1, 8193, 1602516, 68711380, 1289414505, 14350108521, 111239118928, 660994932816, 3202860761145, 13202860761145, 47725572905076, 154718778284148, 457593884876401, 1251308658130545, 3197503726489920

Offset: 0

Bruno Berselli, Oct 05 2010 - Oct 18 2010

This form of recurrence is a general property of the array in A103438 (sums of the first n-th powers).

Bruno Berselli, Table of n, a(n) for n = 0..10000.
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A010801.
Sequences of the form Sum_{j=0..n} j^m : A000217 (m=1), A000330 (m=2), A000537 (m=3), A000538 (m=4), A000539 (m=5), A000540 (m=6), A000541 (m=7), A000542 (m=8), A007487 (m=9), A023002 (m=10), A123095 (m=11), A123094 (m=12), A181134 (m=13).
Cf. A215083, A253712.

[(&+[j^13: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 21 2021

A181134 := proc(n) (bernoulli(14,n+1) - bernoulli(14))/14 ; end proc: seq(A181134(n), n=0..10); # R. J. Mathar, Oct 14 2010

Accumulate[Range[0,20]^13] (* Harvey P. Dale, Oct 30 2017 *)

A181134_list, m = [0], [6227020800, -37362124800, 97037740800, -142702560000, 130456085760, -76592355840, 28805736960, -6711344640, 901020120, -60780720, 1569750, -8190, 1, 0 , 0]
for _ in range(10**2):
    for i in range(14):
        m[i+1]+= m[i]
    A181134_list.append(m[-1]) # Chai Wah Wu, Nov 06 2014

[(bernoulli_polynomial(n+1, 14) - bernoulli(14))/14  for n in (0..30)] # G. C. Greubel, Jul 21 2021

For n>0, a(n) = n*A123094(n) - Sum_{i=0..n-1} A123094(i), where Sum_{i=0..n-1} A123094(i) = A253712(n-1) = (n-1)*n^2*(n+1)*(30*n^10 - 425*n^8 + 2578*n^6 - 8147*n^4 + 12874*n^2 - 7601)/5460.
a(n) = a(-n-1) = (n*(n + 1))^2*(30*n^10 + 150*n^9 + 125*n^8 - 400*n^7 - 326*n^6 + 1052*n^5 + 367*n^4 - 1786*n^3 + 202*n^2 + 1382*n - 691)/420.
G.f.: see comment of Vladeta Jovovic in A000538.
a(n) = -Sum_{j=1..13} j*Stirling1(n+1,n+1-j)*Stirling2(n+13-j,n). - Mircea Merca, Jan 25 2014

A320288 a(n) = n! * [x^n] exp(exp(x)(exp(nx) - 1)/(exp(x) - 1) - n).

Original entry on oeis.org

1, 1, 14, 504, 35054, 4004100, 680823583, 161337142848, 50830272555828, 20549783554154775, 10370522690234157175, 6390016526512315766520, 4721172172018812127424546, 4119920939845363203406535407, 4192465334819134111336349480680, 4920767556196547768620408273728000

Offset: 0

Ilya Gutkovskiy, Oct 09 2018

nonn

G. C. Greubel, Table of n, a(n) for n = 0..218

Cf. A103438, A319508.

Table[n! SeriesCoefficient[Exp[Exp[x] (Exp[n x] - 1)/(Exp[x] - 1) - n], {x, 0, n}], {n, 0, 15}]

a(n)={my(A=O(x^(n+2))); n!*polcoef((exp(exp(x + A)*(exp(n*x + A) - 1)/(exp(x + A) - 1) - n)), n)}; \\ Andrew Howroyd, Nov 04 2018

a(n) = n! * [x^n] exp(exp(x) + exp(2*x) + exp(3*x) + ... + exp(n*x) - n).

a(n) ~ c * exp(n*exp(1) - 3*n) * n^(2*n), where c = exp((exp(1) - 1)/2) / sqrt(exp(1) - 1) = 1.801245710492990660565773944914841332489711300610532... - Vaclav Kotesovec, Jul 02 2022, updated Mar 18 2024

A065551 Triangle of Faulhaber numbers (numerators) read by rows.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, -1, 1, 0, -3, 3, -1, 1, 0, 5, -5, 17, -2, 1, 0, -691, 691, -118, 41, -5, 1, 0, 35, -35, 359, -44, 14, -1, 1, 0, -3617, 3617, -1237, 1519, -293, 22, -7, 1, 0, 43867, -43867, 750167, -13166, 2829, -2258, 217, -4, 1, 0, -1222277, 1222277, -627073, 1540967, -198793, 689, -235, 46, -3, 1

Offset: 0

Wouter Meeussen, Dec 02 2001

From Wolfdieter Lang, Jun 25 2011: (Start)
In the Gessel and Viennot reference f(n,k) = a(n,k)/A065553(n,k), n>=0, k>=0.
(n+1)*f(n,k) = A(n+1,n-k), with Knuth's A(m,k) =
A093556(m,k)/A093557(m,k). See the Knuth reference given in A093556, and the W. Lang link. (End)

			Triangle begins:
{1},
{0, 1},
{0, -1, 1},
{0, 1, -1, 1},
{0, -3, 3, -1, 1},
{0, 5, -5, 17, -2, 1}.

Ira M. Gessel and X. G. Viennot, Determinants, paths and plane partitions, 1989, p. 27, eqn 12.2

Cf. A065553.

Cf. A103438.

sum(n>=0, k>=0, f(n, k)*t^k*x^(2*n+1)/(2*n+1)! ) is the expansion of (cosh(sqrt(1+4*t)*x/2)-cosh(x/2))/t/sinh(x/2).

a(n,k)=numerator(f(n,k)).

A080779 Triangle read by rows: n-th row gives expansion of the series for HarmonicNumber(n, -r).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 0, 6, 12, 6, -4, 0, 40, 60, 24, 0, -60, 0, 300, 360, 120, 120, 0, -840, 0, 2520, 2520, 720, 0, 3360, 0, -11760, 0, 23520, 20160, 5040, -12096, 0, 80640, 0, -169344, 0, 241920, 181440, 40320, 0, -544320, 0, 1814400, 0, -2540160, 0, 2721600, 1814400, 362880

Offset: 0

Wouter Meeussen, Mar 11 2003

The harmonic numbers as used here are defined: HarmonicNumber(n, r) = Zeta(r) - HurwitzZeta(r, n + 1). - Peter Luschny, Mar 21 2024

			The triangle t(n, m) begins:
n\m  0    1    2      3    4      5     6    7 ...
0:   1
1:   1    1
2:   1    3    2
3:   0    6   12      6
4:  -4    0   40     60   24
5:   0  -60    0    300  360    120
6: 120    0 -840      0 2520   2520   720
7:   0 3360    0 -11760    0  23520 20160 5040
...
Row n=8: -12096    0 80640      0 -169344 0 241920 181440 40320;
Row n=9: 0 -544320 0 1814400 0 -2540160 0 2721600 1814400 362880;
Row n=10: 3024000 0 -19958400 0 39916800 0 -39916800 0 33264000 19958400 3628800.
... Reformatted and extended. - _Wolfdieter Lang_, Feb 04 2016

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, pp. 370 - 371.

G. C. Greubel, Rows n=0..100 of triangle, flattened
Wikipedia, Faulhaber's formula

Cf. A264388, A264389, A103438.

RowPoly := n -> local j; n!*add(binomial(n + 1, j) * bernoulli(j, 1) * x^(n - j), j = 0..n): seq(lprint(seq(coeff(RowPoly(n), x, k), k = 0..n)), n = 0..8);
# Peter Luschny, Mar 21 2024

Table[(n+1)! CoefficientList[Sum[k^n, {k, 0, m}]/m, m], {n,1,12}] and for n=0: 1.
a = Join[{{1}}, Table[CoefficientList[Expand[n!*(BernoulliB[n + 1, x + 1] - BernoulliB[n + 1])/x], x], {n, 1, 10}]] Flatten[a] (* Roger L. Bagula and N. J. A. Sloane, Feb 18 2008 *)
T[n_, k_] := Coefficient[ 1/x Integrate[ BernoulliB[n, x + 1], x], x, k]; (* Michael Somos, Aug 18 2018 *)

Row sums are (n + 1)!, last element in row n is n!
Alternative description using Bernoulli polynomials: Let p[x,n]=Sum[k^n,{k,1,x}]; 1/x /. NSolve[p[x,n]-Zeta[n]==0,x] where n>=2. Then t(n,m) = CoefficientList[Expand[n!*(BernoulliB[n + 1, x + 1] - BernoulliB[n + 1])/x], x]. - Roger L. Bagula and N. J. A. Sloane, Feb 18 2008
From Wolfdieter Lang, Feb 04 2016: (Start)
The row polynomials R(n, x) = (n+1)!*F(n, x)/x with F(n,x) = (Sum_{k=1..m} k^n)|{m=x} satisfy the recurrence R(n, x) = n!*(((x + 1)^(n+1) - 1)/x - Sum{k=0..n-1} (binomial(n+1, k)*R(k, x)/(k+1)!)), n >= 1, and R(0, x) = 1. See the Silverman reference, pp. 370 - 371, for F(n, x).
t(n, m) = [x^m] ((Bernoulli(n+1, x+1) - Bernoulli(n+1, 1))/x). See a comment above. For these Bernoulli polynomials see A264388 and A264389. (End)
t(n, m) = t(n-1, m-1) * n/(m+1). - Michael Somos, Aug 18 2018
T(n, k) = [x^k] n!*Sum_{j=0..n} binomial(n+1, j)*Bernoulli(j, 1)*x^(n - j). - Peter Luschny, Mar 21 2024

A142071 Expansion of the exponential generating function 1 - log(1 - x*(exp(z) - 1)), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 7, 12, 6, 0, 1, 15, 50, 60, 24, 0, 1, 31, 180, 390, 360, 120, 0, 1, 63, 602, 2100, 3360, 2520, 720, 0, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040, 0, 1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320, 0, 1, 511, 18660

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

Row n gives the coefficients which express the sums of the n-th powers of the integers as a linear combination of binomial coefficients, thus:
     Sum_{k=1..r} k^n = A103438(n+r,r) = Sum_{k=0..n} T(n+1,k) * C(r,k),
  where, by convention, C(r,k) = 0 whenever r < k. - Robert B Fowler, Jan 16 2023

			Triangle begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   3,    2;
  0, 1,   7,   12,     6;
  0, 1,  15,   50,    60,    24;
  0, 1,  31,  180,   390,   360,   120;
  0, 1,  63,  602,  2100,  3360,  2520,   720;
  0, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040;
  ...

Peter Luschny, Row 0 to 44, recomputed Jan 19 2019

Column k = 0 is A000007.
Cf. A028246, A163626, A000629 (row sums).
Cf. A103438, A007318 (binomial coefficients).

CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
A142071row := proc(n) 1 - log(1 - x*(exp(z) - 1)):
series(%, z, 12): CL(n!*coeff(%, z, n), x) end:
for n from 0 by 1 to 7 do A142071row(n) od;
# Alternative:
A142071Row := proc(n) if n=0 then [1] else
CL(convert(series(polylog(-n+1, z/(1+z)), z, n*2), polynom), z) fi end:
seq(A142071Row(n), n=0..6); # Peter Luschny, Sep 06 2018

T[n_, k_] := If[k==0, Floor[1/(n + 1)], (k - 1)!*StirlingS2[n, k]]; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Jan 06 2024 *)

Row n gives the coefficients of the polynomial defined by p(x, 0) = 1 and for n > 0 p(x, n) = Sum_{k >= 0} k^(n-1)*(x/(1 + x))^k = PolyLog(-n+1, x/(1+x)).

T(n, k) = (k - 1)! * Stirling2(n, k) for k > 0. - Detlef Meya, Jan 06 2024

Edited, T(0,0) = 1 prepended and new name by Peter Luschny, Sep 06 2018

A192001 Triangle with sums of nonnegative integer powers of positive first n integers in the columns.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 5, 1, 5, 10, 14, 9, 1, 6, 15, 30, 36, 17, 1, 7, 21, 55, 100, 98, 33, 1, 8, 28, 91, 225, 354, 276, 65, 1, 9, 36, 140, 441, 979, 1300, 794, 129, 1, 10, 45, 204, 784, 2275, 4425, 4890, 2316, 257, 1

Offset: 1

Wolfdieter Lang, Jun 25 2011

This is the Abramowitz-Stegun table on p. 813, call it s(m,n), with an extra column n=0 with values n added, and read by antidiagonals. a(n,m) = s(n-m,m), n+1 >= m >= 0.
O.g.f. for column no. m >= 0: (x^(m+1)/(1-x)^(m+2))*E(m;x) with the row polynomials E(m;x) = Sum_{p=0..m} A173018(m,p)*x^p of the Eulerian number triangle (proof via the Worpitzky identity). See the Graham et al. reference p. 253-8 for Eulerian numbers, and the Worpitzky identity (6.37) on p. 255.
E.g.f. for diagonals (starting with k=0 for the main diagonal): g(k,x) = exp(x)*(exp((k+1)*x)-1)/(1-exp(x)).
  Compare with (7.77) on p. 353 of the Graham et al. reference.
O.g.f. for diagonals (starting with k=0 for the main diagonal): G(k,z) =(Psi(1/z+1)-Psi(1/z-k-1))/z - 1.
  with the digamma function Psi(z):=(log(Gamma(z)))'.
  Compare with Graham et al., p. 352, eq.(7.76), where H_z=Psi(z+1)+gamma, with the Euler-Mascheroni constant gamma.
The diagonal sequences are, for k=0..9: A000012, A000051, A001550-A001557.
The negative k-diagonal, -a(k+m+1,m), yields the Sheffer z-sequence Shz(k+1;m) for the Sheffer arrays |S1|(k+1) defined in a comment to A094646.
See also A196837 with a W. Lang link, where the o.g.f.s for the diagonals, numbered with k >= 1, are given as G(k,x) = Sum_{m=0..k} (k-m)*S1(k+1,k+1-m)*x^m / Product_{j=1..k} (1-j*x), with S1 the Stirling numbers of the first kind, A048994. - Wolfdieter Lang, Nov 01 2011

			The triangle a(n,m) begins:
n\m    0   1   2    3    4     5     6     7    8   9  10 ...
n=1:   1
n=2:   2   1
n=3:   3   3   1
n=4:   4   6   5    1
n=5:   5  10  14    9    1
n=6:   6  15  30   36   17     1
n=7:   7  21  55  100   98    33     1
n=8:   8  28  91  225  354   276    65     1
n=9:   9  36 140  441  979  1300   794   129    1
n=10: 10  45 204  784 2275  4425  4890  2316  257   1
n=11: 11  55 285 1296 4676 12201 20515 18700 6818 513   1
...  Reformatted and extended by _Wolfdieter Lang_, Jan 12 2013
a(4,2)= 5 = s(2,2) = 1^2 + 2^2.
Recurrence: 55 = a(7,2) = (7-2)*a(6,1) - (a(2,1) + a(3,1) + a(4,1) + a(5,1)) = 5*15 - (1 + 3 + 6 + 10) = 55. - _Wolfdieter Lang_, Jan 12 2013
The first column, m=0 holds the integers 1,2,3,..., equal to the sums of 0th powers of the n first integers. The second column is 1, 1+2, 1+2+3, ... = A000217. The third column are the sums of squares, 1^2, 1^2+2^2, 1^2+2^2+3^3, ... = A000330, etc. - _M. F. Hasler_, Jan 13 2013

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available, see the link), p. 813.
Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1991.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p. 813.

Cf. A103438 (omitting the first column of zeros, reversed).

Flatten[ Table[ HarmonicNumber[-m + n, -m], {n, 1, 10}, {m, 0, n - 1}]] (* Jean-François Alcover, Sep 26 2011 *)

A192001(n,m) = sum(k=1,n-m,k^m) \\ - M. F. Hasler, Jan 13 2013

a(n,m) = s(n-m,m), n-1 >= m >= 0, n >= 1, else 0, with s(n,m) := Sum_{k=1..n} k^m.
O.g.f. column no. m: see a comment above.
O.g.f.s and e.g.f.s for diagonals k >= 0: see a comment above.
Recurrence known by Ibn al-Haytham (see a comment and link under A000537, and comments by Bruno Berselli under the A-numbers of the first column sequences):
  a(n,m) = (n-m)*a(n-1,m-1) - Sum_{j=m..n-2} a(j,m-1), n >= 1, n-1 >= m >= 1. a(n,0) = n. - Wolfdieter Lang, Jan 12 2013

A368479 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} 2^j * j^k.

Original entry on oeis.org

1, 0, 3, 0, 2, 7, 0, 2, 10, 15, 0, 2, 18, 34, 31, 0, 2, 34, 90, 98, 63, 0, 2, 66, 250, 346, 258, 127, 0, 2, 130, 714, 1274, 1146, 642, 255, 0, 2, 258, 2074, 4810, 5274, 3450, 1538, 511, 0, 2, 514, 6090, 18458, 24810, 19098, 9722, 3586, 1023

Offset: 0

Seiichi Manyama, Dec 26 2023

			Square array begins:
    1,   0,    0,     0,      0,      0,       0, ...
    3,   2,    2,     2,      2,      2,       2, ...
    7,  10,   18,    34,     66,    130,     258, ...
   15,  34,   90,   250,    714,   2074,    6090, ...
   31,  98,  346,  1274,   4810,  18458,   71626, ...
   63, 258, 1146,  5274,  24810, 118458,  571626, ...
  127, 642, 3450, 19098, 107754, 616122, 3557610, ...

OEIS Wiki, Eulerian polynomials.

Columns k=0..3 give A126646, A036799, A036800, A036827.
Main diagonal gives A368466.
Cf. A103438, A173018, A368486.

PARI
```
T(n, k) = sum(j=0, n, 2^j*j^k);
```

G.f. of column k: 2*x*A_k(2*x)/((1-x) * (1-2*x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.

A113776 Let S(m, n):=Sum_{k=1..n} k^m; we define a(1):=S(1, 1), a(2):=S(2, S(1, 2)) etc., a(n):=S(n, S(n-1, S(n-2, ...S(1, n)...))).

Original entry on oeis.org

1, 14, 91, 17522596, 1026148893131813787580921710841512880680110612545

Offset: 0

Hieronymus Fischer, Jan 19 2006

nonn

These are iterated decreasing power sums. They are in some way a generalization of iterated triangle sums.

			a(1) = S(1,1) = 1.
a(2) = S(2,S(1,2)) = S(2,3) = 14.
a(3) = S(3,S(2,S(1,3))) = S(3,S(2,6)) = S(3,91) = 17522596.

Cf. A103438 (S(m, n)).

See A115242 for iterated increasing power sums.

Edited by N. J. A. Sloane, Jun 15 2009

A115242 Sequence of iterated increasing power sums. These sums are a generalization of iterated triangle sums.

Original entry on oeis.org

1, 15, 131325321, 210449029945495266124836624698709415244045199778210143675

Offset: 1

Hieronymus Fischer, Jan 19 2006

nonn

These are iterated increasing power sums. These sums are a generalization of iterated triangle sums.

			a(1) = S(1,1) = 1.
a(2) = S(1,S(2,2)) = S(1,5) = 15.
a(3) = S(1,S(2,S(3,3))) = S(1,S(2,36)) = S(1,16206) = 131325321.

Cf. A103438 (S(m, n)).

See A113776 for iterated decreasing power sums.

Let S(m, n) = Sum_{k=1..n} k^m; we define a(1):=S(1, 1), a(2):=S(1, S(2, 2)) etc., a(n):=S(1, S(2, S(3, ...S(n, n)...))).

A322265 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = numerator of Sum_{j=1..n} 1/j^k.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 5, 11, 4, 1, 9, 49, 25, 5, 1, 17, 251, 205, 137, 6, 1, 33, 1393, 2035, 5269, 49, 7, 1, 65, 8051, 22369, 256103, 5369, 363, 8, 1, 129, 47449, 257875, 14001361, 28567, 266681, 761, 9, 1, 257, 282251, 3037465, 806108207, 14011361, 9822481, 1077749, 7129, 10

Offset: 1

Ilya Gutkovskiy, Dec 01 2018

			Square array begins:
  1,       1,          1,              1,                  1,  ...
  2,     3/2,        5/4,            9/8,              17/16,  ...
  3,    11/6,      49/36,        251/216,          1393/1296,  ...
  4,   25/12,    205/144,      2035/1728,        22369/20736,  ...
  5,  137/60,  5269/3600,  256103/216000,  14001361/12960000,  ...

Eric Weisstein's World of Mathematics, Harmonic Number

Columns k=0..10 give A000027, A001008, A007406, A007408, A007410, A099828, A103345, A103347, A103349, A103351, A103716.
Denominators are in A322266.
Cf. A103438, A276485.

Table[Function[k, Numerator[Sum[1/j^k, {j, 1, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
Table[Function[k, Numerator[HarmonicNumber[n, k]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
Table[Function[k, Numerator[SeriesCoefficient[PolyLog[k, x]/(1 - x), {x, 0, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten

G.f. of column k: PolyLog(k,x)/(1 - x), where PolyLog() is the polylogarithm function (for rationals Sum_{j=1..n} 1/j^k).

cp's OEIS Frontend

A181134 Sum of 13th powers: a(n) = Sum_{j=0..n} j^13.

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A320288 a(n) = n! * [x^n] exp(exp(x)*(exp(n*x) - 1)/(exp(x) - 1) - n).

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A065551 Triangle of Faulhaber numbers (numerators) read by rows.

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A080779 Triangle read by rows: n-th row gives expansion of the series for HarmonicNumber(n, -r).

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A142071 Expansion of the exponential generating function 1 - log(1 - x*(exp(z) - 1)), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

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A192001 Triangle with sums of nonnegative integer powers of positive first n integers in the columns.

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A368479 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} 2^j * j^k.

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A320288 a(n) = n! * [x^n] exp(exp(x)(exp(nx) - 1)/(exp(x) - 1) - n).