A304336 -id:A304336 - cp's OEIS frontend

A304338 Row sums of A304336.

1, 1, 4, 26, 239, 2902, 44441, 830636, 18495910, 481474188, 14432543299, 492063896964, 18885525411110, 808850019798316, 38368738864146619, 2002743853356179552, 114374154959904537521, 7110312727864509410026, 479017371580348640009295

Offset: 0

Peter Luschny, May 11 2018

nonn

Cf. A304330, A304334, A304336.

A304338 := n -> add(add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k)/(k!)^2, k=0..n): seq(A304338(n), n=0..18);

a(n) = sum(k=0, n, sum(j=0, k, (-1)^j*binomial(2*k,j)*(k-j)^(2*n)) / (k!)^2); \\ Michel Marcus, May 11 2018

a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(2*k,j)*(k-j)^(2*n) / (k!)^2.

A304330 T(n, k) = Sum_{j=0..k} (-1)^jbinomial(2k, j)(k - j)^(2n), triangle read by rows, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 12, 0, 1, 60, 360, 0, 1, 252, 5040, 20160, 0, 1, 1020, 52920, 604800, 1814400, 0, 1, 4092, 506880, 12640320, 99792000, 239500800, 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600, 0, 1, 65532, 42653520, 3952428480, 111567456000, 1264085222400, 6102480384000, 10461394944000

Offset: 0

Peter Luschny, May 11 2018

			Triangle starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1,    12;
  [3] 0, 1,    60,     360;
  [4] 0, 1,   252,    5040,     20160;
  [5] 0, 1,  1020,   52920,    604800,    1814400;
  [6] 0, 1,  4092,  506880,  12640320,   99792000,   239500800;
  [7] 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600;

José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024.

Row sums are A100872, T(n,2) = A058896, T(n,n) = A002674, T(n,n-1)= A091032.

Cf. A304334, A304336.

T := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k):
for n from 0 to 8 do seq(T(n, k), k=0..n) od;

T(n, k) = sum(j=0, k, (-1)^j*binomial(2*k, j)*(k - j)^(2*n)); \\ Michel Marcus, Aug 03 2025

A304334 T(n, k) = Sum_{j=0..k} (-1)^jbinomial(2k, j)(k - j)^(2n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 6, 0, 1, 30, 60, 0, 1, 126, 840, 840, 0, 1, 510, 8820, 25200, 15120, 0, 1, 2046, 84480, 526680, 831600, 332640, 0, 1, 8190, 780780, 9609600, 30270240, 30270240, 8648640, 0, 1, 32766, 7108920, 164684520, 929728800, 1755673920, 1210809600, 259459200

Offset: 0

Peter Luschny, May 11 2018

			Triangle starts:
[0] 1
[1] 0, 1
[2] 0, 1,     6
[3] 0, 1,    30,      60
[4] 0, 1,   126,     840,       840
[5] 0, 1,   510,    8820,     25200,     15120
[6] 0, 1,  2046,   84480,    526680,    831600,     332640
[7] 0, 1,  8190,  780780,   9609600,  30270240,   30270240,    8648640
[8] 0, 1, 32766, 7108920, 164684520, 929728800, 1755673920, 1210809600, 259459200

Row sums are bisection of A081562, T(n,n) ~ A000407, T(n,n-1) ~ A048854(n,2), T(n,2) ~ A002446.

Cf. A304330, A304336.

A304334 := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k)/k!:
for n from 0 to 8 do seq(A304334(n, k), k=0..n) od;

T(n, k) = sum(j=0, k, (-1)^j*binomial(2*k, j)*(k - j)^(2*n))/k!;
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 11 2018

T(n, k) = A304330(n, k) / k!.

A303675 Triangle read by rows: coefficients in the sum of odd powers as expressed by Faulhaber's theorem, T(n, k) for n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 6, 1, 120, 30, 1, 5040, 1680, 126, 1, 362880, 151200, 17640, 510, 1, 39916800, 19958400, 3160080, 168960, 2046, 1, 6227020800, 3632428800, 726485760, 57657600, 1561560, 8190, 1, 1307674368000, 871782912000, 210680870400, 22313491200, 988107120, 14217840, 32766, 1

Offset: 1

Kolosov Petro, May 08 2018

T(n,k) are the coefficients in an identity due to Faulhaber: Sum_{j=0..n} j^(2*m-1) = Sum_{k=1..m} T(m,k) binomial(n+k, 2*k). See the Knuth reference, page 10.

More explicitly, Faulhaber's theorem asserts that, given integers n >= 0, m >= 1 and odd, Sum_{k=1..n} k^m = Sum_{k=1..(m+1)/2} C(n+k,n-k)*[(1/k)*Sum_{j=0..k-1} (-1)^j*C(2*k,j)*(k-j)^(m+1)]. The coefficients T(m, k) are indicated by square brackets. Sums similar to this inner part are A304330, A304334, A304336; however, these triangles are (0,0)-based and lead to equivalent but slightly more systematic representations. - Peter Luschny, May 12 2018

			The triangle begins (see the Knuth reference p. 10):
         1;
         6,          1;
       120,         30,         1;
      5040,       1680,       126,        1;
    362880,     151200,     17640,      510,       1;
  39916800,   19958400,   3160080,   168960,    2046,    1;
6227020800, 3632428800, 726485760, 57657600, 1561560, 8190, 1;
.
Let S(n, m) = Sum_{j=1..n} j^m. Faulhaber's formula gives for m = 7 (m odd!):
F(n, 7) = 5040*C(n+4, 8) + 1680*C(n+3, 6) + 126*C(n+2, 4) + C(n+1, 2).
Faulhaber's theorem asserts that for all n >= 1 S(n, 7) = F(n, 7).
If n = 43 the common value is 1600620805036.

John H. Conway and Richard Guy, The Book of Numbers, Springer (1996), p. 107.

Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:9207222 [math.CA], 1992.
Petro Kolosov, Polynomial identities involving central factorial numbers, GitHub, 2024. See pp. 3, 6.

First column is a bisection of A000142, second column is a bisection of A001720.
Row sums give A100868.
Cf. A008955, A008957, A036969 and A304330, A304334, A304336.

T := proc(n,k) local m; m := n-k;
2*(2*m+1)!*add((-1)^(j+m)*(j+1)^(2*n)/((j+m+2)!*(m-j)!), j=0..m) end:
seq(seq(T(n, k), k=1..n), n=1..8); # Peter Luschny, May 09 2018

(* After Peter Luschny's above formula. *)
T[n_, k_] := (1/(n-k+1))*Sum[(-1)^j*Binomial[2*(n-k+1), j]*((n-k+1) - j)^(2*n), {j, 0, n-k+1}]; Column[Table[T[n, k], {n, 1, 10}, {k, 1, n}], Center]

def A303675(n, k): return factorial(2*(n-k)+1)*A008957(n, k)
for n in (1..7): print([A303675(n, k) for k in (1..n)]) # Peter Luschny, May 10 2018

T(n, k) = (2*(n-k)+1)!*A008957(n, k), n >= 1, 1 <= k <= n.

T(n, k) = (1/m)*Sum_{j=0..m} (-1)^j*binomial(2*m,j)*(m-j)^(2*n) where m = n-k+1. - Peter Luschny, May 09 2018

New name by Peter Luschny, May 10 2018

A385567 Triangle read by rows: T(n,k) is the numerator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2n) Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

Original entry on oeis.org

1, 1, 1, 1, 0, -1, 1, -1, 0, 1, 1, -4, 2, 0, -1, 1, -5, 3, -3, 0, 5, 1, -4, 17, -10, 5, 0, -691, 1, -35, 287, -118, 691, -691, 0, 7, 1, -8, 112, -352, 718, -280, 140, 0, -3617, 1, -21, 66, -293, 4557, -3711, 10851, -10851, 0, 43867, 1, -40, 217, -4516, 2829, -26332, 750167, -438670, 219335, 0, -174611

Offset: 0

Kolosov Petro, Jul 31 2025

The companion triangle with the denominators is A386728.

Extension of A093556 with k in the range 0 <= k <= n, and n >= 0.

			Triangle begins:
---------------------------------------------------------------------------------
k =   0    1     2     3     4       5       6        7       8      9      10
---------------------------------------------------------------------------------
n=0:  1;
n=1:  1,   1;
n=2:  1,   0,  -1;
n=3:  1,  -1,   0,     1;
n=4:  1,  -4,   2,     0,   -1;
n=5:  1,  -5,   3,    -3,    0,      5;
n=6:  1,  -4,  17,   -10,    5,      0,   -691;
n=7:  1, -35, 287,  -118,  691,   -691,      0,       7;
n=8:  1,  -8, 112,  -352,  718,   -280,    140,       0,  -3617;
n=9:  1, -21,  66,  -293, 4557,  -3711,  10851,  -10851,      0, 43867;
n=10: 1, -40, 217, -4516, 2829, -26332, 750167, -438670, 219335,     0, -174611;
...

Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:9207222 [math.CA], 1992, see page 16.
Petro Kolosov, Faulhaber's coefficients: Examples, GitHub, 2025.
Petro Kolosov, Mathematica programs, GitHub, 2025.

Cf. A386728 (denominators).
Cf. A303675, A304330, A304334, A304336.
Cf. A093558/A093559, A335951/A335952, A093556/A093557.

FaulhaberCoefficient[n_, k_] := 0;
FaulhaberCoefficient[n_, k_] := (-1)^(n - k) * Sum[Binomial[2 n, n - k - j]* Binomial[n - k + j, j] * (n - k - j)/(n - k + j) * BernoulliB[n + k + j], {j, 0, n - k}] /; 0 <= k < n;
FaulhaberCoefficient[n_, k_] := BernoulliB[2 n] /; k == n;
Flatten[Table[Numerator[FaulhaberCoefficient[n, k]], {n, 0, 10}, {k, 0, n}]]

T(n,k) = numerator(if (k==n, bernfrac(2*n), if (kMichel Marcus, Aug 03 2025

A(n,k) = 0 if k>n or n<0
A(n,k) = (-1)^(n - k) * Sum_{j=0..n-k} binomial(2n, n - k - j) * binomial(n - k + j, j) * (n - k - j)/(n - k + j) * B_{n + k + j}, if 0 <= k < n;
A(n,k) = B_{2n}, if k = n;
T(n,k) = numerator(A(n,k)).

A386728 Triangle read by rows: T(n,k) is the denominator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2n) Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

Original entry on oeis.org

1, 1, 6, 1, 1, 30, 1, 2, 1, 42, 1, 3, 3, 1, 30, 1, 2, 1, 2, 1, 66, 1, 1, 2, 1, 1, 1, 2730, 1, 6, 15, 3, 15, 30, 1, 6, 1, 1, 3, 3, 3, 1, 1, 1, 510, 1, 2, 1, 1, 5, 2, 5, 10, 1, 798, 1, 3, 2, 7, 1, 3, 42, 21, 21, 1, 330, 1, 2, 3, 2, 1, 6, 15, 3, 5, 10, 1, 138, 1

Offset: 0

Kolosov Petro, Jul 31 2025

The companion triangle with the numerators is A385567.

Extension of A093557 with k in the range 0 <= k <= n.

			Triangle begins:
  ---------------------------------------------------------
  k =   0  1   2   3    4    5    6   7    8    9    10
  ---------------------------------------------------------
  n=0:  1;
  n=1:  1, 6;
  n=2:  1, 1, 30;
  n=3:  1, 2,  1, 42;
  n=4:  1, 3,  3,  1, 30;
  n=5:  1, 2,  1,  2,  1, 66;
  n=6:  1, 1,  2,  1,  1,  1, 2730;
  n=7:  1, 6, 15,  3, 15, 30,    1,  6;
  n=8:  1, 1,  3,  3,  3,  1,    1,  1, 510;
  n=9:  1, 2,  1,  1,  5,  2,    5, 10,   1, 798;
  n=10: 1, 3,  2,  7,  1,  3,   42, 21,  21,   1, 330;
  ...

Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:9207222 [math.CA], 1992, see page 16.
Petro Kolosov, Faulhaber's coefficients: Examples, GitHub, 2025.
Petro Kolosov, Mathematica programs, GitHub, 2025.

Cf. A385567 (numerators).
Cf. A303675, A304330, A304334, A304336.
Cf. A093558/A093559, A335951/A335952, A093556/A093557.

FaulhaberCoefficient[n_, k_] := 0;
FaulhaberCoefficient[n_, k_] := (-1)^(n - k) * Sum[Binomial[2 n, n - k - j]* Binomial[n - k + j, j] * (n - k - j)/(n - k + j) * BernoulliB[n + k + j], {j, 0, n - k}] /; 0 <= k < n;
FaulhaberCoefficient[n_, k_] := BernoulliB[2 n] /; k == n;
Flatten[Table[Denominator[FaulhaberCoefficient[n, k]], {n, 0, 10}, {k, 0, n}]]

T(n,k) = denominator(if (k==n, bernfrac(2*n), if (kMichel Marcus, Aug 03 2025

A(n,k) = 0 if k>n or n<0;
A(n,k) = (-1)^(n - k) * Sum_{j=0..n-k} binomial(2n, n - k - j) * binomial(n - k + j, j) * (n - k - j)/(n - k + j) * B_{n + k + j}, if 0 <= k < n;
A(n,k) = B_{2n}, if k = n;
T(n,k) = denominator(A(n,k)).

cp's OEIS Frontend

A304338 Row sums of A304336.

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A304330 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n.

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A304334 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.

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A303675 Triangle read by rows: coefficients in the sum of odd powers as expressed by Faulhaber's theorem, T(n, k) for n >= 1, 1 <= k <= n.

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A385567 Triangle read by rows: T(n,k) is the numerator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2*n) * Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

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A386728 Triangle read by rows: T(n,k) is the denominator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2*n) * Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

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A304330 T(n, k) = Sum_{j=0..k} (-1)^jbinomial(2k, j)(k - j)^(2n), triangle read by rows, n >= 0 and 0 <= k <= n.

A304334 T(n, k) = Sum_{j=0..k} (-1)^jbinomial(2k, j)(k - j)^(2n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.

A385567 Triangle read by rows: T(n,k) is the numerator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2n) Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

A386728 Triangle read by rows: T(n,k) is the denominator of A(n,k), such that A(n,k) satisfies the identity for sums of odd powers: Sum_{k=1..p} k^(2n-1) = 1/(2n) Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.