A008955 -id:A008955 - cp's OEIS frontend

A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2n, 2k) (A036969).

1, -1, 1, 4, -5, 1, -36, 49, -14, 1, 576, -820, 273, -30, 1, -14400, 21076, -7645, 1023, -55, 1, 518400, -773136, 296296, -44473, 3003, -91, 1, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1, 1625702400, -2483133696, 1017067024, -173721912, 14739153, -669188, 16422, -204, 1

Offset: 1

M. F. Hasler, Feb 03 2012

This is a signed version of A008955 with rows in reverse order. - Peter Luschny, Feb 04 2012

			Triangle starts:
  [1]         1;
  [2]        -1,        1;
  [3]         4,       -5,         1;
  [4]       -36,       49,       -14,       1;
  [5]       576,     -820,       273,     -30,       1;
  [6]    -14400,    21076,     -7645,    1023,     -55,    1;
  [7]    518400,  -773136,    296296,  -44473,    3003,  -91,    1;
  [8] -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1;

José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 6.
Mark W. Coffey and Matthew C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.

Cf. A036969, A008955, A008275, A121408, A001044 (column 1), A101686 (alternating row sums), A234324 (central terms).

# From Peter Luschny, Feb 29 2024: (Start)
ogf := n -> local j; z^2*mul(z^2 - j^2, j = 1..n-1):
Trow := n -> local k; seq(coeff(expand(ogf(n)), z, 2*k), k = 1..n):
# Alternative:
f := w -> (w^sqrt(t) + w^(-sqrt(t)))/2: egf := f((x/2 + sqrt(1 + (x/2)^2))^2):
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, k), k = 1..n):  # (End)
# Assuming offset 0:
rowpoly := n -> (-1)^n * pochhammer(1 - sqrt(x), n) * pochhammer(1 + sqrt(x), n):
row := n -> local k; seq(coeff(expand(rowpoly(n)), x, k), k = 0..n):
seq(print(row(n)), n = 0..7);  # Peter Luschny, Aug 03 2024

rows = 10;
t[n_, k_] := 2*Sum[j^(2*n)*(-1)^(k - j)/((k - j)!*(k + j)!), {j, 1, k}];
T = Table[t[n, k], {n, 1, rows}, {k, 1, rows}] // Inverse;
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018 *)

select(concat(Vec(matrix(10,10,n,k,T(n,k)/*from A036969*/)~^-1)), x->x)

def A204579(n, k): return (-1)^(n-k)*A008955(n, n-k)
for n in (0..7): print([A204579(n, k) for k in (0..n)]) # Peter Luschny, Feb 05 2012

T(n, k) = (-1)^(n-k)*A008955(n, n-k). - Peter Luschny, Feb 05 2012
T(n, k) = Sum_{i=k-n..n-k} (-1)^(n-k+i)*s(n,k+i)*s(n,k-i) = Sum_{i=0..2*k} (-1)^(n+i)*s(n,i)*s(n,2*k-i), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 07 2012
From Peter Bala, Aug 29 2012: (Start)
T(n, k) = T(n-1, k-1) - (n-1)^2*T(n-1, k). (Recurrence equation.)
Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/{(2*n)!/2^n} and
L(x) = 2*{arcsinh(sqrt(x/2))}^2 = Sum_{n >=1} (-1)^n*(n-1)!^2*x^n/{(2*n)!/2^n}.
L(x) is the compositional inverse of E(x) - 1.
A generating function for the triangle is E(t*L(x)) = 1 + t*x + t*(-1 + t)*x^2/6 + t*(4 - 5*t + t^2)*x^3/90 + ..., where the sequence of denominators [1,1,6,90,...] is given by (2*n)!/2^n. Cf. A008275 with generating function exp(t*log(1+x)).
The e.g.f. is E(t*L(x^2/2)) = cosh(2*sqrt(t)*arcsinh(x/2)) = 1 + t*x^2/2! + t*(t-1)*x^4/4! + t*(t-1)*(t-4)*x^6/6! + .... (End)
From Peter Luschny, Feb 29 2024: (Start)
T(n, k) = [z^(2*k)] z^2*Product_{j=1..n-1} (z^2 - j^2).
T(n, k) = (2*n)! * [t^k] [x^(2*n)] (w^sqrt(t) + w^(-sqrt(t)))/2 where w = (x/2 + sqrt(1 + (x/2)^2))^2. (End)
T(n, k) = [x^k] (-1)^n * Pochhammer(1 - sqrt(x), n) * Pochhammer(1 + sqrt(x), n), assuming offset 0. - Peter Luschny, Aug 03 2024
Integral_{0..oo} x^s / (cosh(x))^(2*n) dx = (2^(2*n - s - 1) * s! * (-1)^(n-1)) / (2*n - 1)!)*Sum_{k=1..n} T(n,k)*DirichletEta(s - 2*k + 2). - Ammar Khatab, Apr 11 2025

Typo in data corrected by Peter Luschny, Feb 05 2012

A249677 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^3*x).

Original entry on oeis.org

1, 1, 1, 1, 9, 8, 1, 36, 251, 216, 1, 100, 2555, 16280, 13824, 1, 225, 15055, 335655, 2048824, 1728000, 1, 441, 63655, 3587535, 74550304, 444273984, 373248000, 1, 784, 214918, 25421200, 1305074809, 26015028256, 152759224512, 128024064000, 1, 1296, 616326, 135459216, 14320729209, 694213330464, 13472453691584, 78340747014144, 65548320768000

Offset: 0

Paul D. Hanna, Nov 03 2014

Column 1 forms the squares of the triangular numbers (A000537).
Main diagonal forms the cubes of the factorial numbers (A000442).
Row sums equal Product_{k=1..n} (k^3 + 1) = n!*Product_{k=1..n} (k*(k-1) + 1) = n!*A130032(n).

			Triangle begins:
  1;
  1, 1;
  1, 9, 8;
  1, 36, 251, 216;
  1, 100, 2555, 16280, 13824;
  1, 225, 15055, 335655, 2048824, 1728000;
  1, 441, 63655, 3587535, 74550304, 444273984, 373248000;
  1, 784, 214918, 25421200, 1305074809, 26015028256, 152759224512, 128024064000;
  1, 1296, 616326, 135459216, 14320729209, 694213330464, 13472453691584, 78340747014144, 65548320768000; ...

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A000537, A000442, A008955, A107415, A130032.

{T(n,k)=polcoeff(prod(m=0,n,1 + m^3*x +x*O(x^n)),k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

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

A370707 Triangle read by rows: T(n, k) = (-1)^kProduct_{j=0..k-1} (j - n)(j + n), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 12, 1, 9, 72, 360, 1, 16, 240, 2880, 20160, 1, 25, 600, 12600, 201600, 1814400, 1, 36, 1260, 40320, 1088640, 21772800, 239500800, 1, 49, 2352, 105840, 4233600, 139708800, 3353011200, 43589145600, 1, 64, 4032, 241920, 13305600, 638668800, 24908083200, 697426329600, 10461394944000

Offset: 0

Peter Luschny, Feb 27 2024

The definition, and also the representation T(n, k) = ff(n, k) * rf(n, k) (see the first formula), makes it natural to call this triangle the central factorial numbers.

			Triangle starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  4,   12;
  [3] 1,  9,   72,    360;
  [4] 1, 16,  240,   2880,   20160;
  [5] 1, 25,  600,  12600,  201600,   1814400;
  [6] 1, 36, 1260,  40320, 1088640,  21772800,  239500800;
  [7] 1, 49, 2352, 105840, 4233600, 139708800, 3353011200, 43589145600;
.
T(n, k) is a product where 'n' is the 'center' and 'k' is the 'half-length' of the product. For instance, T(5, 4) = (5-3)*(5-2)*(5-1)*5 * 5*(5+1)*(5+2)*(5+3) = 201600. Now consider the polynomial P(4, x) = -36*x^2 + 49*x^4 - 14*x^6 + x^8. Evaluating this polynomial at x = 5 shows P(4, 5) = 201600 = T(5, 4). The coefficients of the polynomial are row 4 of A269944.

Diagonals: A002674, A327882.
Columns: A000290, A047928.
Cf. A370704 (row sums), A370706, A094728, A048994 (Stirling1), A130595 (order 0), A269947 (order 3)

T := (n, k) -> local j; (-1)^k * mul((j - n)*(j + n), j = 0..k-1):
seq(seq(T(n, k), k = 0..n), n = 0..8);
# The central factorial numbers:
cf := (n, k) -> ifelse(k = 0, 1, n*(n + k - 1)! / (n - k)! ):
for n from 0 to 6 do seq(cf(n, k), k = 0..n) od;
# Alternative (recurrence):
T := proc(n, k) option remember;
if k = 0 then 1 else T(n, k - 1)*(n^2 - (k - 1)^2) fi end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;
# Illustrating the connection with the cf-polynomials and their coefficients:
cfpoly := (n,x) -> local k; mul(x^2 - k^2, k = 0..n-1):
A370707row := n -> local k; [seq(cfpoly(k, n), k = 0..n)]:
A204579row := n -> local k; [seq(coeff(cfpoly(n, x), x, 2*k), k = 0..n)]:
for n from 0 to 5 do lprint([n], A370707row(n), A204579row(n)) od;

T[n_, k_] := If[n == 0, 1, -n Pochhammer[1 - n - k, 2 k - 1]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

from math import prod
def T(n, k): return (-1)**k * prod((j - n)*(j + n) for j in range(k))
print([T(n, k) for n in range(8) for k in range(n + 1)])

def T(n, k): return falling_factorial(n, k) * rising_factorial(n, k)
for n in range(9): print([T(n, k) for k in range(n + 1)])

T(n, k) = FallingFactorial(n, k) * RisingFactorial(n, k).
T(n, k) = (n*(n + k - 1)!)/(n - k)! if k > 0, and T(n, 0) = 1.
Calling the numbers in the second formula cf leads to the memorable form cf(n, k) = ff(n, k) * rf(n, k). This identity generalizes to the function
  cf(x, n) = x*Gamma(x + n)/Gamma(x - n + 1) for n > 0 and cf(x, 0) = 1.
The last equation shows that the variable 'n' does not have to be an integer but can be any complex number if only the quotient remains defined (which one often can achieve by taking the limit). Indeed, in the classical Steffensen-Riordan case, n/2 is used instead of n, which leads to the complex situation Sloane discusses in A008955.
T(n, k) = -n*Pochhammer(1 - n - k, 2*k - 1) for n > 0.
T(n, k) = k!*binomial(n, k)*Pochhammer(n, k) = k!*A370706(n, k).
T(n, n) = n!*Pochhammer(n, n) (valid for n >= 0, whereas T(n, n) = (2*n)!/2 = A002674(n) is valid for n >= 1 only).
T(n, k) = T(n, k - 1)*(n^2 - (k - 1)^2) if k > 0, otherwise 1. (Recurrence)
The cf(n, k) are values of the polynomials Pcf(n, x) = Product_{k=0..n-1} (x^2 - k^2), whose coefficients vanish for odd powers and for even powers are A269944.
T(n, k) = Pcf(k, n) where Pcf(k,x) = Sum_{j=0..k} (-1)^(k-j)*A269944(k,j)*x^(2*j).
The central factorials can be described in three different ways: By the product T(n, k) = f(n, k) * rf(n, k), by the complex function cf(x, n), and through the polynomials Pcf(n, x). Although these relations are self-contained, they are regarded as only one-half of a more general notion, namely as central factorials of the first kind.
There is a fundamental connection with the Stirling numbers of first kind (A048994). The easiest way to see this is to generalize the definition: Let CF(z, s) = Product_{j=0..n-1} (z - s(j)), where s(j) is some complex sequence. Then the coefficients of CF(z, s) are equal to the Stirling_1 numbers if s = 0, 1, 2, ..., n, ..., and they are equal to the coefficients of our Pcf(n, z) polynomials if s = 0, 1, 4, ..., n^2, .... (This is also why A269944 is called the 'Stirling cycle numbers of order 2'. For completeness, if s = 1, 1, 1, ..., then the coefficients of CF(z, s), the 'Stirling cycle numbers of order 0', are the signed Pascal triangle A130595. See A269947 for order 3.)

A107415 Triangle, read by rows: T(0,0) = 1; T(n,k) = n!*T(n-1,k) - T(n-1,k-1).

Original entry on oeis.org

1, 1, -1, 2, -3, 1, 12, -20, 9, -1, 288, -492, 236, -33, 1, 34560, -59328, 28812, -4196, 153, -1, 24883200, -42750720, 20803968, -3049932, 114356, -873, 1, 125411328000, -215488512000, 104894749440, -15392461248, 579404172, -4514276, 5913, -1

Offset: 0

Gerald McGarvey, May 26 2005

For n>0, the row sums are 0. For n>1, sum(k=0..n) 2^k*T(n,k) = 0. The first subdiagonal (1,-3,9,-33,...) is an alternating signed version of A007489 (sum of k!, k=1..n). The first column is A000178 (superfactorials).

Also triangle of coefficients in expansion of Product_{k=0..n} (k! - x) in ascending powers of x. - Seiichi Manyama, Sep 24 2021

			Triangle begins
         1;
         1,        -1;
         2,        -3,        1;
        12,       -20,        9,       -1;
       288,      -492,      236,      -33,      1;
     34560,    -59328,    28812,    -4196,    153,   -1;
  24883200, -42750720, 20803968, -3049932, 114356, -873, 1;
(1 - x) * (2 - x) = 2 - 3*x + x^2, (1 - x) * (2 - x) * (6 - x) = 12 - 20*x + 9*x^2 - x^3, etc. - _Seiichi Manyama_, Sep 24 2021

Seiichi Manyama, Rows n = 0..43, flattened

Cf. A000178, A008955, A249677.

t(n, k) = {if (k < 0, return (0)); if (n < k, return (0)); if (n == 0, return (1)); return (n!*t(n-1, k) - t(n-1, k-1));} \\ Michel Marcus, Apr 11 2013

row(n) = Vecrev(prod(k=1, n, k!-x)); \\ Seiichi Manyama, Sep 24 2021

A348014 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^k*x).

Original entry on oeis.org

1, 1, 1, 1, 5, 4, 1, 32, 139, 108, 1, 288, 8331, 35692, 27648, 1, 3413, 908331, 26070067, 111565148, 86400000, 1, 50069, 160145259, 42405161203, 1216436611100, 5205269945088, 4031078400000

Offset: 0

Seiichi Manyama, Sep 24 2021

			Triangle begins:
  1;
  1,    1;
  1,    5,      4;
  1,   32,    139,      108;
  1,  288,   8331,    35692,     27648;
  1, 3413, 908331, 26070067, 111565148, 86400000;

Seiichi Manyama, Rows n = 0..37, flattened

Column k=1 gives A001923.
The diagonal of the triangle is A002109.
Cf. A007318, A008955, A023531, A108084, A173007, A173008, A249677.

T(n, k) = if(k==0, 1, if(k==n, prod(j=1, n, j^j), T(n-1, k)+n^n*T(n-1, k-1)));

PARI
```
row(n) = Vecrev(prod(k=1, n, 1+k^k*x));
```

T(0,0) = 1; T(n,k) = T(n-1,k) + n^n * T(n-1,k-1).

A370705 Triangle read by rows: T(n, k) = numerator(CF(n, k)) where CF(n, k) = n! * [x^k] [t^n] (t/2 + sqrt(1 + (t/2)^2))^(2*x).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 9, 0, -5, 0, 1, 0, 0, 4, 0, -5, 0, 1, 0, -225, 0, 259, 0, -35, 0, 1, 0, 0, -36, 0, 49, 0, -14, 0, 1, 0, 11025, 0, -3229, 0, 987, 0, -21, 0, 1, 0, 0, 576, 0, -820, 0, 273, 0, -30, 0, 1

Offset: 0

Peter Luschny, Mar 02 2024

The rational triangle R(n, k) contains the central factorial numbers. The central factorial of the first kind is the even subtriangle of R, while the central factorial of the second kind is the odd subtriangle. Since the terms of the even subtriangle can be seen as integers, the rational nature of these numbers is generally disregarded. The denominators of the central factorial of second kind are powers of 4; therefore, they are often studied as integers in the form 4^(n-k)*R(n, k). We will refer to the subtriangles by CF1(n, k) and CF2(n, k).
We recall that if T(n, k) is a number triangle (0 <= k <= n) then
   Teven(n, k) = [T(n, k), k=0..n step 2), n=0..len step 2]
   is the even subtriangle of T, and the odd subtriangle of T is
   Todd(n, k) = [T(n, k), k=1..n step 2), n=1..len step 2], where
   'k=a..o step s' denotes the subrange [a, a+s, a+2*s, ..., a+s*floor((o-a)/s)].
The central factorial numbers have their origins in approximation theory and numerical mathematics. They were undoubtedly used for a long time when J. F. Steffensen used them to construct quadrature formulas and presented them in 1924 at the 7th ICM. Four decades later, Carlitz and Riordan adopted the idea for use in combinatorics. While Steffensen originally referred to the numbers as "central differences of nothing," the second part of the name was later omitted.

			Triangle starts:
[0] 1;
[1] 0,     1;
[2] 0,     0,   1;
[3] 0,    -1,   0,     1;
[4] 0,     0,  -1,     0,  1;
[5] 0,     9,   0,    -5,  0,   1;
[6] 0,     0,   4,     0, -5,   0,   1;
[7] 0,  -225,   0,   259,  0, -35,   0,   1;
[8] 0,     0, -36,     0, 49,   0, -14,   0, 1;
[9] 0, 11025,   0, -3229,  0, 987,   0, -21, 0, 1;

Johan Frederik Steffensen, On a class of quadrature formulae. Proceedings of the International Mathematical Congress Toronto 1924, Vol 2, pp. 837-844.

P. L. Butzer, M. Schmidt, E. L. Stark and L. Vogt. Central factorial numbers; their main properties and some applications, Num. Funct. Anal. Optim., 10 (1989) 419-488.
Leonard Carlitz and John Riordan, The Divided Central Differences of Zero, Canadian Journal of Mathematics, Volume 15, 1963, pp. 94-100.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
Johan Frederik Steffensen, On the Definition of the Central Factorial, Journal of the Institute of Actuaries, Volume 64, Issue 2, July 1933, pp. 165-168.

See the discussion by Sloane in A008955 of Riordan's notation. In particular, the notation 'T' below does not refer to the present triangle.
Central factorials (rational, general case): (this triangle) / A370703;
t(2n, 2k) (first kind, 'even case') A204579; (signed, T(n, 0) missing)
|t(2n, 2k)|                         A269944; (unsigned, T(n, 0) = 0^n)
|t(2n, 2n-2k)|                      A008955;
|t(2n+1, 2n+1-2k)|*4^k              A008956;
T(2n, 2k) (second kind, 'odd case') A269945, A036969;
T(2n+1, 2k+1)*4^(n-k)               A160562.

gf := (t/2 + sqrt(1 + (t/2)^2))^(2*x): ser := series(gf, t, 20):
ct := n -> n!*coeff(ser, t, n):
T := (n, k) -> numer(coeff(ct(n), x, k)):
seq(seq(T(n, k), k = 0..n), n = 0..10);
# Filtering the central factorials of the first resp. second kind:
CF1 := (T,len) -> local n,k; seq(print(seq(T(n,k), k=0..n, 2)), n = 0..len, 2);
CF2 := (T,len) -> local n,k; seq(print(seq(T(n,k), k=1..n, 2)), n = 1..len, 2);

cp's OEIS Frontend

A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2*n, 2*k) (A036969).

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A249677 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^3*x).

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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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A370707 Triangle read by rows: T(n, k) = (-1)^k*Product_{j=0..k-1} (j - n)*(j + n), for 0 <= k <= n.

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A107415 Triangle, read by rows: T(0,0) = 1; T(n,k) = n!*T(n-1,k) - T(n-1,k-1).

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A348014 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^k*x).

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A370705 Triangle read by rows: T(n, k) = numerator(CF(n, k)) where CF(n, k) = n! * [x^k] [t^n] (t/2 + sqrt(1 + (t/2)^2))^(2*x).

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A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2n, 2k) (A036969).

A370707 Triangle read by rows: T(n, k) = (-1)^kProduct_{j=0..k-1} (j - n)(j + n), for 0 <= k <= n.