Kolosov Petro - cp's OEIS frontend

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.

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)).

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)).

A374452 Iterated rascal triangle R3: T(n,k) = Sum_{m=0..3} binomial(n-k,m)*binomial(k,m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 69, 56, 28, 8, 1, 1, 9, 36, 84, 121, 121, 84, 36, 9, 1, 1, 10, 45, 120, 195, 226, 195, 120, 45, 10, 1

Offset: 0

Kolosov Petro, Jul 08 2024

Triangle T(n,k) is the third triangle R3 among the rascal-family triangles; A077028 is triangle R1, A374378 is triangle R2.
Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 7).
Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 3).

			Triangle begins:
--------------------------------------------------
k=     0   1   2   3    4    5    6   7   8   9 10
--------------------------------------------------
n=0:   1
n=1:   1   1
n=2:   1   2   1
n=3:   1   3   3   1
n=4:   1   4   6   4    1
n=5:   1   5  10  10    5    1
n=6:   1   6  15  20   15    6    1
n=7:   1   7  21  35   35   21    7   1
n=8:   1   8  28  56   69   56   28   8   1
n=9:   1   9  36  84  121  121   84  36   9   1
n=10:  1  10  45 120  195  226  195  120  45  10  1

Jena Gregory, Brandt Kronholm and Jacob White, Iterated rascal triangles, Aequationes mathematicae, 2023.
Jena Gregory, Iterated rascal triangles, Theses and Dissertations. 1050., The University of Texas Rio Grande Valley, 2022.
Amelia Gibbs and Brian K. Miceli, Two Combinatorial Interpretations of Rascal Numbers, arXiv:2405.11045 [math.CO], 2024.
Philip K. Hotchkiss, Student Inquiry and the Rascal Triangle, arXiv:1907.07749 [math.HO], 2019.
Philip K. Hotchkiss, Generalized Rascal Triangles, Journal of Integer Sequences, Vol. 23, 2020.
Petro Kolosov, Identities in Iterated Rascal Triangles, 2024.

Cf. A077028, A000027, A000217, A000292, A005894, A247608, A008860, A374378, A007318, A096943, A096940

t[n_, k_] := Sum[Binomial[n - k, m]*Binomial[k, m], {m, 0, 3}]; Column[Table[t[n, k], {n, 0, 12}, {k, 0, n}], Left]

T(n,k) = 1 + k*(n-k) + 1/4*(k-1)*k*(n-k-1)*(n-k) + 1/36*(k-2)*(k-1)*k*(n-k-2)*(n-k-1)*(n-k).
Row sums give A008860(n).
Diagonal T(n+1, n) gives A000027(n).
Diagonal T(n+2, n) gives A000217(n).
Diagonal T(n+3, n) gives A000292(n).
Diagonal T(n+4, n) gives A005894(n).
Diagonal T(n+6, n) gives A247608(n).
Column k=4 difference binomial(n+8, 4) - T(n+8, 4) gives C(n+4,4)=A007318(n+4,4).
Column k=5 difference binomial(n+9, 5) - T(n+9, 5) gives sixth column of (1,5)-Pascal triangle A096943.
G.f.: (1 + 4*x^6*y^3 - 3*x*(1 + y) - 6*x^5*y^2*(1 + y) + 2*x^4*y*(2 + 7*y+ 2*y^2) + x^2*(3 + 10*y + 3*y^2) - x^3*(1 + 11*y + 11*y^2 + y^3))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Jul 09 2024

A374378 Iterated rascal triangle R2: T(n,k) = Sum_{m=0..2} binomial(n-k,m)*binomial(k,m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 19, 15, 6, 1, 1, 7, 21, 31, 31, 21, 7, 1, 1, 8, 28, 46, 53, 46, 28, 8, 1, 1, 9, 36, 64, 81, 81, 64, 36, 9, 1, 1, 10, 45, 85, 115, 126, 115, 85, 45, 10, 1, 1, 11, 55, 109, 155, 181, 181, 155, 109, 55, 11, 1

Offset: 0

Kolosov Petro, Jul 06 2024

Triangle T(n,k) is the second triangle R2 among the rascal-family triangles; A374452 is triangle R3; A077028 is triangle R1.
Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 5).
Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 2).

			Triangle begins:
--------------------------------------------------
k=     0   1   2   3    4    5    6   7   8   9 10
--------------------------------------------------
n=0:   1
n=1:   1   1
n=2:   1   2   1
n=3:   1   3   3   1
n=4:   1   4   6   4    1
n=5:   1   5  10  10    5    1
n=6:   1   6  15  19   15    6    1
n=7:   1   7  21  31   31   21    7   1
n=8:   1   8  28  46   53   46   28   8   1
n=9:   1   9  36  64   81   81   64  36   9   1
n=10:  1  10  45  85  115  126  115  85  45  10  1

Kolosov Petro, Table of n, a(n) for n = 0..1325
Amelia Gibbs and Brian K. Miceli, Two Combinatorial Interpretations of Rascal Numbers, arXiv:2405.11045 [math.CO], 2024.
Jena Gregory, Brandt Kronholm, and Jacob White, Iterated rascal triangles, Aequationes mathematicae, 2023.
Jena Gregory, Iterated rascal triangles, Theses and Dissertations. 1050., The University of Texas Rio Grande Valley, 2022.
Philip K. Hotchkiss, Student Inquiry and the Rascal Triangle, arXiv:1907.07749 [math.HO], 2019.
Philip K. Hotchkiss, Generalized Rascal Triangles, Journal of Integer Sequences, Vol. 23, 2020.
Petro Kolosov, Identities in Iterated Rascal Triangles, 2024.

Cf. A077028, A007318, A000027, A000217, A005448, A056108, A212656, A000292, A095667, A095666, A006261.

t[n_, k_]:=Sum[Binomial[n - k, m]*Binomial[k, m], {m, 0, 2}]; Column[Table[t[n, k], {n, 0, 12}, {k, 0, n}], Center]

T(n,k) = 1 + k*(n-k) + (1/4)*(k-1)*k*(n-k-1)*(n-k).
Row sums give A006261(n).
Diagonal T(n+1, n) gives A000027(n).
Diagonal T(n+2, n) gives A000217(n).
Diagonal T(n+3, n) gives A005448(n).
Diagonal T(n+4, n) gives A056108(n).
Diagonal T(n+5, n) gives A212656(n).
Column k=3 difference binomial(n+6, 3) - T(n+6, 3) gives C(n+3,3)=A007318(n+3,3).
Column k=4 difference binomial(n+7, 4) - T(n+7, 4) gives fifth column of (1,4)-Pascal triangle A095667.
G.f.: (1 + 3*x^4*y^2 - (2*x + 3*x^3*y)*(1 + y) + x^2*(1 + 5*y + y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 09 2024

A306548 Triangle T(n,k) read by rows, where the k-th column is the shifted self-convolution of the power function n^k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 10, 8, 1, 0, 0, 5, 20, 34, 16, 1, 0, 0, 6, 35, 104, 118, 32, 1, 0, 0, 7, 56, 259, 560, 418, 64, 1, 0, 0, 8, 84, 560, 2003, 3104, 1510, 128, 1, 0, 0, 9, 120, 1092, 5888, 16003, 17600, 5554, 256, 1, 0, 0, 10, 165, 1968, 14988, 64064, 130835, 101504, 20758, 512, 1, 0, 0

Offset: 0

Kolosov Petro, Feb 23 2019

For n > 0 an odd-power identity n^(2m+1)+1, m >= 0 can be found using the current sequence. The sum of the n-th diagonal of T(n,k) over 0 <= k <= m multiplied by A(m,k) gives n^(2m+1)-1, where A(m,k) = A302971(m,k)/A304042(m,k). For example, consider the case n=4, m=2: the n-th diagonal of T(n, 0 <= k <= m) is {5, 10, 34}, and the m-th row of triangle A(m, 0 <= k <= m) is {1, 0, 30}, thus (3+1)^5 + 1 = 5*1 + 10*0 + 34*30 = 1025.

			==================================================================
k=    0     1     2     3      4      5     6    7    8    9    10
==================================================================
n=0:  2;
n=1:  2,    0;
n=2:  3,    0,    0;
n=3:  4,    1,    0,    0;
n=4:  5,    4,    1,    0,     0;
n=5:  6,   10,    8,    1,     0,     0;
n=6:  7,   20,   34,   16,     1,     0,    0;
n=7:  8,   35,  104,  118,    32,     1,    0,   0;
n=8:  9,   56,  259,  560,   418,    64,    1,   0,   0;
n=9:  10,  84,  560, 2003,  3104,  1510,  128,   1,   0,   0;
n=10: 11, 120, 1092, 5888, 16003, 17600, 5554, 256,   1,   0;   0;
...

D. V. Widder et al., The Convolution Transform, Bull. Amer. Math. Soc. 60 (1954), 444-456.
Wikipedia, Convolution.
Wikipedia, Convolution power.

Nonzero terms of columns k=0..5 give: A000027, A000292, A033455, A145216, A145217, A145218.
Partial sums of columns k=1..2 give: A000332, A259181.
Cf. A302971, A304042, A198633, A000079.

f[m_, s_] := Piecewise[{{s^m, s >= 0}, {0, True}}];
F[n_, m_] := Sum[f[m, n - k]*f[m, k], {k, -Infinity, +Infinity}];
T[n_, k_] := F[n - k, k];
Column[Table[T[n, k], {n, 0, 12}, {k, 0, n}], Left]

f(m, s) = s^m, if s >= 0;
f(m, s) = 0, otherwise.
F(n,m) = Sum_{k} f(m, n-k) * f(m, k), -oo < k < +oo;
T(n,k) = F(n-k, k).

Edited by Kolosov Petro, Mar 13 2019

A320047 Consider coefficients U(m,l,k) defined by the identity Sum_{k=1..l} Sum_{j=0..m} A302971(m,j)/A304042(m,j) * k^j * (T-k)^j = Sum_{k=0..m} (-1)^(m-k) * U(m,l,k) * T^k that holds for all positive integers l,m,T. This sequence gives 2-column table read by rows, where n-th row lists coefficients U(1,n,k) for k = 0, 1 and n >= 1.

Original entry on oeis.org

5, 6, 28, 18, 81, 36, 176, 60, 325, 90, 540, 126, 833, 168, 1216, 216, 1701, 270, 2300, 330, 3025, 396, 3888, 468, 4901, 546, 6076, 630, 7425, 720, 8960, 816, 10693, 918, 12636, 1026, 14801, 1140, 17200, 1260, 19845, 1386, 22748, 1518, 25921, 1656

Offset: 1

Kolosov Petro, Oct 04 2018

For l=T, the identity takes the form T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k, which holds for all positive integers T and m.

			        column  column
   l      k=0     k=1
  ---   ------  ------
   1       5       6
   2      28      18
   3      81      36
   4     176      60
   5     325      90
   6     540     126
   7     833     168
   8    1216     216
   9    1701     270
  10    2300     330
  11    3025     396
  12    3888     468
  ...

Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, More details on derivation of present sequence.
Petro Kolosov, Mathematica program, verifies the identity T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k for m=0,1,...,12.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.

The case m=2 is A316349.
The case m=3 is A316387.
Column k=0 is A275709.
Column k=1 is A028896.
Cf. A302971, A304042, A287326, A300656, A300785.

(* Define the R[n,k] := A302971(n,k)/A304042(n,k) *)
R[n_, k_] := 0
R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
   Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
   BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
(* Define the U(m,l,t) coefficients *)
U[m_, l_, t_] := (-1)^m Sum[Sum[Binomial[j, t] R[m,j] k^(2 j - t) (-1)^j, {j, t, m}], {k, 1, l}];
(* Define the value of the variable 'm' to be m = 1 for A320047 *)
m = 1;
(* Print first 10 rows of U(m,l,t) coefficients for 'm' defined above *)
Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]

U(m,l,t) = (-1)^m * Sum_{k=1..l} Sum_{j=t..m} binomial(j,t) * R(m,j) * k^{2j-t} * (-1)^j, where m = 1, l >= 1 and R(m,j) = A302971(m,j)/A304042(m,j); after Max Alekseyev, see links.
Conjectures from Colin Barker, Aug 03 2019: (Start)
G.f.: x*(5 + 6*x + 8*x^2 - 6*x^3 - x^4) / ((1 - x)^4*(1 + x)^4).
a(n) = (4 - 4*(-1)^n - 3*(-5+(-1)^n)*n - 3*(-3+(-1)^n)*n^2 + (1+(-1)^(1+n))*n^3) / 8.
a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n>8.
(End)

A316349 Consider coefficients U(m,L,k) defined by the identity Sum_{k=1..L} Sum_{j=0..m} A302971(m,j)/A304042(m,j) * k^j * (T-k)^j = Sum_{k=0..m} (-1)^(m-k) * U(m,L,k) * T^k that holds for all positive integers L,m,T. This sequence gives 3-column table read by rows, where the n-th row lists coefficients U(2,n,k) for k = 0, 1, 2; n >= 1.

Original entry on oeis.org

31, 60, 30, 512, 540, 150, 2943, 2160, 420, 10624, 6000, 900, 29375, 13500, 1650, 68256, 26460, 2730, 140287, 47040, 4200, 263168, 77760, 6120, 459999, 121500, 8550, 760000, 181500, 11550, 1199231, 261360, 15180, 1821312, 365040, 19500, 2678143, 496860, 24570, 3830624, 661500, 30450

Offset: 1

Kolosov Petro, Jun 29 2018

For L=T, the identity takes form T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k, which holds for all positive integers T and m.

			       column   column  column
   L     k=0      k=1     k=2
  --  -------  -------  ------
   1       31       60      30
   2      512      540     150
   3     2943     2160     420
   4    10624     6000     900
   5    29375    13500    1650
   6    68256    26460    2730
   7   140287    47040    4200
   8   263168    77760    6120
   9   459999   121500    8550
  10   760000   181500   11550
  11  1199231   261360   15180
  12  1821312   365040   19500
  ...

Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, More details on derivation of present sequence.
Petro Kolosov, Mathematica program, verifies the identity T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k for m=0,1,...,12.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.

The case m=1 is A320047.
The case m=3 is A316387.
Column k=0 is A316457.
Column k=1 is A316458.
Column k=2 is A316459.
Cf. A302971, A304042, A287326, A300656, A300785.

(* Define the R[n,k] := A302971(m,j)/A304042(m,j) *)
R[n_, k_] := 0
R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
   Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
   BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
(* Define the U(m,l,t) coefficients *)
U[m_, l_, t_] := (-1)^m Sum[Sum[Binomial[j, t] R[m,j] k^(2 j - t) (-1)^j, {j, t, m}], {k, 1, l}];
(* Define the value of the variable 'm', should be m = 2 for A316349 *)
m = 2;
(* Print first 10 rows of U(m,l,t) coefficients over l: 1 <= l <= 10 *)
Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]

U(2,n,0) = 6*n^5 + 15*n^4 + 10*n^3; U(2,n,1) = 15*n^4 + 30*n^3 + 15*n^2; U(2,n,2) = 10*n^3 + 15*n^2 + 5*n. - Max Alekseyev, Sep 06 2018
From Colin Barker, Jul 06 2018: (Start)
G.f.: x*(31 + 60*x + 30*x^2 + 326*x^3 + 180*x^4 - 30*x^5 + 336*x^6 - 180*x^7 - 30*x^8 + 26*x^9 - 60*x^10 + 30*x^11 + x^12) / ((1 - x)^6*(1 + x + x^2)^6).
a(n) = 6*a(n-3) - 15*a(n-6) + 20*a(n-9) - 15*a(n-12) + 6*a(n-15) - a(n-18) for n>18. (End)
U(m,L,t) = (-1)^m * Sum_{k=1..L} Sum_{j=t..m} binomial(j,t) * R(m,j) * k^{2j-t} * (-1)^j, where m = 1, L >= 1 and R(m,j) = A302971(m,j)/A304042(m,j); after Max Alekseyev, see links.

Edited by Max Alekseyev, Sep 06 2018

A316387 Consider coefficients U(m,L,k) defined by the identity Sum_{k=1..L} Sum_{j=0..m} A302971(m,j)/A304042(m,j) * k^j * (T-k)^j = Sum_{k=0..m} (-1)^(m-k) * U(m,L,k) * T^k that holds for all positive integers L,m,T. This sequence gives 4-column table read by rows, where the n-th row lists coefficients U(3,n,k) for k = 0, 1, 2, 3; n >= 1.

Original entry on oeis.org

125, 406, 420, 140, 9028, 13818, 7140, 1260, 110961, 115836, 41160, 5040, 684176, 545860, 148680, 14000, 2871325, 1858290, 411180, 31500, 9402660, 5124126, 955500, 61740, 25872833, 12182968, 1963920, 109760, 62572096, 25945416, 3684240, 181440, 136972701, 50745870, 6439860, 283500, 276971300, 92745730, 10639860, 423500

Offset: 1

Kolosov Petro, Jul 01 2018

For T=L, the identity takes form T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k, which holds for all positive integers T and m.

			            column      column      column   column
   L          k=0         k=1         k=2      k=3
  --  ------------  ----------  ----------  -------
   1           125         406         420      140
   2          9028       13818        7140     1260
   3        110961      115836       41160     5040
   4        684176      545860      148680    14000
   5       2871325     1858290      411180    31500
   6       9402660     5124126      955500    61740
   7      25872833    12182968     1963920   109760
   8      62572096    25945416     3684240   181440
   9     136972701    50745870     6439860   283500
  10     276971300    92745730    10639860   423500
  11     524988145   160386996    16789080   609840
  12     943023888   264896268    25498200   851760
  13    1618774781   420839146    37493820  1159340
  14    2672907076   646725030    53628540  1543500
  15    4267591425   965662320    74891040  2016000
  16    6616398080  1406064016   102416160  2589440
  17    9995653693  2002403718   137494980  3277260
  18   14757360516  2796022026   181584900  4093740
  19   21343778801  3835983340   236319720  5054000
  20   30303773200  5179983060   303519720  6174000
  ...

Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, More details on derivation of present sequence.
Petro Kolosov, Mathematica program, verifies the identity T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k for m=0,1,...,12.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.

The case m=1 is A320047.
The case m=2 is A316349.
Column k=0 is A317981.
Column k=1 is A317982.
Column k=2 is A317983.
Column k=3 is A317984.
Cf. A287326, A300656, A300785, A302971, A304042.

(* Define the R[n,k] := A302971(m,j)/A304042(m,j) *)
R[n_, k_] := 0
R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
   Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
   BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
(* Define the U(m,l,t) coefficients *)
U[m_, l_, t_] := (-1)^m Sum[Sum[Binomial[j, t] R[m,j] k^(2 j - t) (-1)^j, {j, t, m}], {k, 1, l}];
(* Define the value of the variable 'm' *)
m = 3;
(* Print first 10 rows of U(m,l,t) coefficients for 'm' defined above *)
Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]

U(3,n,0) = 20*n^7 + 70*n^6 + 70*n^5 - 28*n^3 - 7*n^2; U(3,n,1) = 70*n^6 + 210*n^5 + 175*n^4 - 42*n^2 - 7*n; U(3,n,2) = 84*n^5 + 210*n^4 + 140*n^3 - 14*n; U(3,n,3) = 35*n^4 + 70*n^3 + 35*n^2. - Max Alekseyev, Sep 06 2018
From Colin Barker, Jul 09 2018; corrected by Max Alekseyev, Sep 06 2018: (Start)
G.f.: x*(125 + 406*x + 420*x^2 + 140*x^3 + 8028*x^4 + 10570*x^5 + 3780*x^6 + 140*x^7 + 42237*x^8 + 16660*x^9 - 4200*x^10 - 1120*x^11 + 42272*x^12 - 16660*x^13 - 4200*x^14 + 1120*x^15 + 8007*x^16 - 10570*x^17 + 3780*x^18 - 140*x^19 + 132*x^20 - 406*x^21 + 420*x^22 - 140*x^23 - x^24) / ((1 - x)^8*(1 + x)^8*(1 + x^2)^8).
a(n) = 8*a(n-4) - 28*a(n-8) + 56*a(n-12) - 70*a(n-16) + 56*a(n-20) - 28*a(n-24) + 8*a(n-28) - a(n-32) for n>32. (End)
U(m,L,t) = (-1)^m * Sum_{k=1..L} Sum_{j=t..m} binomial(j,t) * R(m,j) * k^{2j-t} * (-1)^j, where m = 1, L >= 1 and R(m,j) = A302971(m,j)/A304042(m,j); after Max Alekseyev, see links. - Kolosov Petro, Oct 04 2018

Edited by Max Alekseyev, Sep 06 2018

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

A304042 Triangle read by rows: T(n,k) is the denominator of R(n,k) defined implicitly by the identity Sum_{i=0..l-1} Sum_{j=0..m} R(m,j)(l-i)^ji^j = l^(2*m+1) holding for all l,m >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Kolosov Petro, May 05 2018

			Triangle begins:
-----------------------------------------------------
k=    0  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15
-----------------------------------------------------
n=0:  1;
n=1:  1, 1;
n=2:  1, 1, 1;
n=3:  1, 1, 1, 1;
n=4:  1, 1, 1, 1, 1;
n=5:  1, 1, 1, 1, 1, 1;
n=6:  1, 1, 1, 1, 1, 1, 1;
n=7:  1, 1, 1, 1, 1, 1, 1, 1;
n=8:  1, 1, 1, 1, 1, 1, 1, 1, 1;
n=9:  1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
n=10: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
n=11: 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1;
n=12: 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1;
n=13: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
n=14: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
n=15: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Antti Karttunen, Table of n, a(n) for n = 0..10439 (the first 144 rows of triangle)
P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.
C. Jordan, Calculus of Finite Differences, Budapest, 1939. [Annotated scans of pages 448-450 only]
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Definition and table of values.
Petro Kolosov, An unusual identity for odd-powers, arXiv:2101.00227 [math.GM], 2021.
Petro Kolosov, 106.37 An unusual identity for odd-powers, The Mathematical Gazette, 2022.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
Petro Kolosov, A novel proof of power rule in calculus, GitHub, 2024. See p. 5.
Petro Kolosov, Odd-power identity via multiplication of certain matrices, GitHub, 2024. See p. 4.
Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.
Petro Kolosov, Discussion on coefficients of odd polynomial identity, GitHub, 2025.
MathOverflow, Discussion of these coefficients, 2018.

Numerators are shown in A302971.

Cf. A287326, A300656, A300785, A007318, A027641, A027642, A055012, A077028, A000146, A002882, A003245, A127187, A127188, A074909, A164555.

R[n_, k_] := 0
R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
   Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
   BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
T[n_, k_] := Denominator[R[n, k]];
(* Print Fifteen Initial rows of Triangle A304042 *)
Column[ Table[ T[n, k], {n, 0, 15}, {k, 0, n}], Center]

up_to = 1274; \\ = binomial(50+1,2)-1
A304042aux(n, k) = if((k<0)||(k>n),0,(k+k+1)*binomial(2*k, k)*if(k==n,1,sum(j=k+k+1,n, A304042aux(n, j)*binomial(j, k+k+1)*((-1)^(j-1))/(j-k)*bernfrac(2*(j-k)))));
A304042tr(n, k) = denominator(A304042aux(n, k));
A304042list(up_to) = { my(v = vector(up_to), i=0); for(n=0,oo, for(k=0,n, if(i++ > up_to, return(v)); v[i] = A304042tr(n,k))); (v); };
v304042 = A304042list(1+up_to);
A304042(n) = v304042[1+n]; \\ Antti Karttunen, Nov 07 2018

Recurrence given by Max Alekseyev (see the MathOverflow link):
R(n, k) = 0 if k < 0 or k > n.
R(n, k) = (2k+1)*binomial(2k, k) if k = n.
R(n, k) = (2k+1)*binomial(2k, k)*Sum_{j=2k+1..n} R(n, j)*binomial(j, 2k+1)*(-1)^(j-1)/(j-k)*Bernoulli(2j-2k), otherwise.
T(n, k) = denominator(R(n, k)).

cp's OEIS Frontend

User: Kolosov Petro

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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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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A374452 Iterated rascal triangle R3: T(n,k) = Sum_{m=0..3} binomial(n-k,m)*binomial(k,m).

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A374378 Iterated rascal triangle R2: T(n,k) = Sum_{m=0..2} binomial(n-k,m)*binomial(k,m).

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A306548 Triangle T(n,k) read by rows, where the k-th column is the shifted self-convolution of the power function n^k, n >= 0, 0 <= k <= n.

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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/(2n) Sum_{k=0..n-1} A(n,k) * (p^2+p)^(n-k), for all integers p >= 1.

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.

A304042 Triangle read by rows: T(n,k) is the denominator of R(n,k) defined implicitly by the identity Sum_{i=0..l-1} Sum_{j=0..m} R(m,j)(l-i)^ji^j = l^(2*m+1) holding for all l,m >= 0.