A269944 -id:A269944 - cp's OEIS frontend

A374848 Obverse convolution A000045**A000045; see Comments.

0, 1, 2, 16, 162, 3600, 147456, 12320100, 2058386904, 701841817600, 488286500625000, 696425232679321600, 2038348954317776486400, 12259459134020160144810000, 151596002479762016373851690400, 3855806813438155578522841251840000

Offset: 0

Clark Kimberling, Jul 31 2024

nonn

The obverse convolution of sequences
  s = (s(0), s(1), ...) and t = (t(0), t(1), ...)
is introduced here as the sequence s**t given by
s**t(n) = (s(0)+t(n)) * (s(1)+t(n-1)) * ... * (s(n)+t(0)).
Swapping * and + in the representation s(0)*t(n) + s(1)*t(n-1) + ... + s(n)*t(0)
of ordinary convolution yields s**t.
If x is an indeterminate or real (or complex) variable, then for every sequence t of real (or complex) numbers, s**t is a sequence of polynomials p(n) in x, and the zeros of p(n) are the numbers -t(0), -t(1), ..., -t(n).
Following are abbreviations in the guide below for triples (s, t, s**t):
F = (0,1,1,2,3,5,...) = A000045, Fibonacci numbers
L = (2,1,3,4,7,11,...) = A000032, Lucas numbers
P = (2,3,5,7,11,...) = A000040, primes
T = (1,3,6,10,15,...) = A000217, triangular numbers
C = (1,2,6,20,70, ...) = A000984, central binomial coefficients
LW = (1,3,4,6,8,9,...) = A000201, lower Wythoff sequence
UW = (2,5,7,10,13,...) = A001950, upper Wythoff sequence
[ ] = floor
In the guide below, sequences s**t are identified with index numbers Axxxxxx; in some cases, s**t and Axxxxxx differ in one or two initial terms.
Table 1. s = A000012 = (1,1,1,1...) = (1);
t = A000012; 1                 s**t = A000079; 2^(n+1)
t = A000027; n                 s**t = A000142; (n+1)!
t = A000040, P                 s**t = A054640
t = A000040, P           (1/3) s**t = A374852
t = A000079, 2^n               s**t = A028361
t = A000079, 2^n         (1/3) s**t = A028362
t = A000045, F                 s**t = A082480
t = A000032, L                 s**t = A374890
t = A000201, LW                s**t = A374860
t = A001950, UW                s**t = A374864
t = A005408, 2*n+1             s**t = A000165, 2^n*n!
t = A016777, 3*n+1             s**t = A008544
t = A016789, 3*n+2             s**t = A032031
t = A000142, n!                s**t = A217757
t = A000051, 2^n+1             s**t = A139486
t = A000225, 2^n-1             s**t = A006125
t = A032766, [3*n/2]           s**t = A111394
t = A034472, 3^n+1             s**t = A153280
t = A024023, 3^n-1             s**t = A047656
t = A000217, T                 s**t = A128814
t = A000984, C                 s**t = A374891
t = A279019, n^2-n             s**t = A130032
t = A004526, 1+[n/2]           s**t = A010551
t = A002264, 1+[n/3]           s**t = A264557
t = A002265, 1+[n/4]           s**t = A264635
Sequences (c)**L, for c=2..4: A374656 to A374661
Sequences (c)**F, for c=2..6: A374662, A374662, A374982 to A374855
The obverse convolutions listed in Table 1 are, trivially, divisibility sequences. Likewise, if s = (-1,-1,-1,...) instead of s = (1,1,1,...), then s**t is a divisibility sequence for every choice of t; e.g. if s = (-1,-1,-1,...) and t = A279019, then s**t = A130031.
Table 2. s = A000027 = (0,1,2,3,4,5,...) = (n);
t = A000027, n                 s**t = A007778, n^(n+1)
t = A000290, n^2               s**t = A374881
t = A000040, P                 s**t = A374853
t = A000045, F                 s**t = A374857
t = A000032, L                 s**t = A374858
t = A000079, 2^n               s**t = A374859
t = A000201, LW                s**t = A374861
t = A005408, 2*n+1             s**t = A000407, (2*n+1)! / n!
t = A016777, 3*n+1             s**t = A113551
t = A016789, 3*n+2             s**t = A374866
t = A000142, n!                s**t = A374871
t = A032766, [3*n/2]           s**t = A374879
t = A000217, T                 s**t = A374892
t = A000984, C                 s**t = A374893
t = A038608, n*(-1)^n          s**t = A374894
Table 3. s = A000290 = (0,1,4,9,16,...) = (n^2);
t = A000290, n^2               s**t = A323540
t = A002522, n^2+1             s**t = A374884
t = A000217, T                 s**t = A374885
t = A000578, n^3               s**t = A374886
t = A000079, 2^n               s**t = A374887
t = A000225, 2^n-1             s**t = A374888
t = A005408, 2*n+1             s**t = A374889
t = A000045, F                 s**t = A374890
Table 4. s = t;
s = t = A000012, 1             s**s = A000079; 2^(n+1)
s = t = A000027, n             s**s = A007778, n^(n+1)
s = t = A000290, n^2           s**s = A323540
s = t = A000045, F             s**s = this sequence
s = t = A000032, L             s**s = A374850
s = t = A000079, 2^n           s**s = A369673
s = t = A000244, 3^n           s**s = A369674
s = t = A000040, P             s**s = A374851
s = t = A000201, LW            s**s = A374862
s = t = A005408, 2*n+1         s**s = A062971
s = t = A016777, 3*n+1         s**s = A374877
s = t = A016789, 3*n+2         s**s = A374878
s = t = A032766, [3*n/2]       s**s = A374880
s = t = A000217, T             s**s = A375050
s = t = A005563, n^2-1         s**s = A375051
s = t = A279019, n^2-n         s**s = A375056
s = t = A002398, n^2+n         s**s = A375058
s = t = A002061, n^2+n+1       s**s = A375059
If n = 2*k+1, then s**s(n) is a square; specifically,
s**s(n) = ((s(0)+s(n))*(s(1)+s(n-1))*...*(s(k)+s(k+1)))^2.
If n = 2*k, then s**s(n) has the form 2*s(k)*m^2, where m is an integer.
Table 5. Others
s = A000201, LW      t = A001950, UW         s**t = A374863
s = A000045, F       t = A000032, L          s**t = A374865
s = A005843, 2*n     t = A005408, 2*n+1      s**t = A085528, (2*n+1)^(n+1)
s = A016777, 3*n+1   t = A016789, 3*n+2      s**t = A091482
s = A005408, 2*n+1   t = A000045, F          s**t = A374867
s = A005408, 2*n+1   t = A000032, L          s**t = A374868
s = A005408, 2*n+1   t = A000079, 2^n        s**t = A374869
s = A000027, n       t = A000142, n!         s**t = A374871
s = A005408, 2*n+1   t = A000142, n!         s**t = A374872
s = A000079, 2^n     t = A000142, n!         s**t = A374874
s = A000142, n!      t = A000045, F          s**t = A374875
s = A000142, n!      t = A000032, L          s**t = A374876
s = A005408, 2*n+1   t = A016777, 3*n+1      s**t = A352601
s = A005408, 2*n+1   t = A016789, 3*n+2      s**t = A064352
Table 6. Arrays of coefficients of s(x)**t(x), where s(x) and t(x) are polynomials
s(x)              t(x)            s(x)**t(x)
 n                 x              A132393
n^2                x              A269944
x+1               x+1             A038220
x+2               x+2             A038244
 x                x+3             A038220
nx                x+1             A094638
 1              x^2+x+1           A336996
n^2 x             x+1             A375041
n^2 x            2x+1             A375042
n^2 x             x+2             A375043
2^n x             x+1             A375044
2^n              2x+1             A375045
2^n              x+2              A375046
x+1              F(n)             A375047
x+1             x+F(n)            A375048
x+F(n)          x+F(n)            A375049

			a(0) = 0 + 0 = 0
a(1) = (0+1) * (1+0) = 1
a(2) = (0+1) * (1+1) * (1+0) = 2
a(3) = (0+2) * (1+1) * (1+1) * (2+0) = 16
As noted above, a(2*k+1) is a square for k>=0. The first 5 squares are 1, 16, 3600, 12320100, 701841817600, with corresponding square roots 1, 4, 60, 3510, 837760.
If n = 2*k, then s**s(n) has the form 2*F(k)*m^2, where m is an integer and F(k) is the k-th Fibonacci number; e.g., a(6) = 2*F(3)*(192)^2.

Cf. A000045, A374850-to-A374881, A374884-to-A374894, A375041-to-A375049.

a:= n-> (F-> mul(F(n-j)+F(j), j=0..n))(combinat[fibonacci]):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 02 2024

s[n_] := Fibonacci[n]; t[n_] := Fibonacci[n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]

a(n)=prod(k=0, n, fibonacci(k) + fibonacci(n-k)) \\ Andrew Howroyd, Jul 31 2024

a(n) ~ c * phi^(3*n^2/4 + n) / 5^((n+1)/2), where c = QPochhammer(-1, 1/phi^2)^2/2 if n is even and c = phi^(1/4) * QPochhammer(-phi, 1/phi^2)^2 / (phi + 1)^2 if n is odd, and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 01 2024

A269945 Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 21, 14, 1, 0, 1, 85, 147, 30, 1, 0, 1, 341, 1408, 627, 55, 1, 0, 1, 1365, 13013, 11440, 2002, 91, 1, 0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1, 0, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1

Offset: 0

Peter Luschny, Mar 22 2016

Also known as central factorial numbers T(2*n, 2*k) (cf. A036969).

The analog for the Stirling cycle numbers is A269944.

			Triangle starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,   1]
  [3] [0, 1,   5,    1]
  [4] [0, 1,  21,   14,   1]
  [5] [0, 1,  85,  147,  30,  1]
  [6] [0, 1, 341, 1408, 627, 55, 1]

M. W. Coffey and M. 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.
Peter Luschny, The P-transform.
Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.

Columns k=0..5 give A000007, A000012, A002450(n-1), A002451(n-3), A383838(n-4), A383840(n-5).
Variants are: A008957, A036969.
Cf. A007318 (order 0), A048993 (order 1), A269948 (order 3).
Cf. A000330 (subdiagonal), A002450 (column 2), A135920 (row sums), A269941, A269944 (Stirling cycle), A298851 (central terms).

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + k^2*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od;
# Alternatively with the P-transform (cf. A269941):
A269945_row := n -> PTrans(n, n->`if`(n=1, 1, 1/(n*(4*n-2))), (n, k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269945_row(n)), n=0..8);
# Using the exponential generating function:
egf := 1 + t^2*(cosh(2*sinh(t*x/2)/t));
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, 2*(n-k+1)), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # Peter Luschny, Feb 29 2024

T[n_, n_] = 1; T[n_ /; n >= 0, k_] /; 0 <= k < n := T[n, k] = T[n - 1, k - 1] + k^2*T[n - 1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Nov 27 2017 *)

# uses[PtransMatrix from A269941]
stirset2 = lambda n: 1 if n == 1 else 1/(n*(4*n-2))
norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stirset2, norm)
for m in M: print(m)

T(n, k) = (-1)^k*((2*n)! / (2*k)!)*P[n, k](s(n)) where P is the P-transform and s(n) = 1/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n, 2) = (4^(n - 1) - 1)/3 for n >= 2 (cf. A002450).
T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
From Fabián Pereyra, Apr 25 2022: (Start)
T(n, k) = (1/(2*k)!)*Sum_{j=0..2*k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n).
T(n, k) = Sum_{j=2*k..2*n} (-k)^(2*n - j)*binomial(2*n, j)*Stirling2(j, 2*k).
T(n, k) = Sum_{j=0..2*n} (-1)^(j - k)*Stirling2(2*n - j, k)*Stirling2(j, k). (End)
T(n, k) = (2*n)! [t^(2*(n-k+1))] [x^(2*n)] (1 + t^2*(cosh(2*sinh(t*x/2)/t))). - Peter Luschny, Feb 29 2024

A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 12, 10, 0, 4, 30, 60, 35, 0, 5, 60, 210, 280, 126, 0, 6, 105, 560, 1260, 1260, 462, 0, 7, 168, 1260, 4200, 6930, 5544, 1716, 0, 8, 252, 2520, 11550, 27720, 36036, 24024, 6435, 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310

Offset: 0

Peter Luschny, Mar 20 2015

The rows give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1) / (n!*(n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 31 2022

This is related to the cluster fans of type B (see Fomin and Zelevinsky reference) - F. Chapoton, Nov 17 2022.

			[1]
[0, 1]
[0, 2,   3]
[0, 3,  12,   10]
[0, 4,  30,   60,   35]
[0, 5,  60,  210,  280,  126]
[0, 6, 105,  560, 1260, 1260,  462]
[0, 7, 168, 1260, 4200, 6930, 5544, 1716]
.
R_0(x) = 1/(x-1)^0.
R_1(x) = 0/(x-1)^1 + 1/(x-1)^2.
R_2(x) = 0/(x-1)^2 + 2/(x-1)^3 + 3/(x-1)^4.
R_3(x) = 0/(x-1)^3 + 3/(x-1)^4 + 12/(x-1)^5 + 10/(x-1)^6.
Then k!*[x^k] R_n(x) is A001286(k+2) and A001754(k+3) for n = 2, 3 respectively.
.
Seen as an array A(n, k) = binomial(n + k, k)*binomial(n + 2*k - 1, n + k):
[0] 1, 1,   3,   10,    35,    126,     462, ...
[1] 0, 2,  12,   60,   280,   1260,    5544, ...
[2] 0, 3,  30,  210,  1260,   6930,   36036, ...
[3] 0, 4,  60,  560,  4200,  27720,  168168, ...
[4] 0, 5, 105, 1260, 11550,  90090,  630630, ...
[5] 0, 6, 168, 2520, 27720, 252252, 2018016, ...
[6] 0, 7, 252, 4620, 60060, 630630, 5717712, ...

Seiichi Manyama, Rows n = 0..139, flattened
F. Chapoton Enumerative Properties of Generalized Associahedra, Sém. Loth. Comb. B51b (2004).
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
S. Fomin and A. Zelevinsky Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3.
Yunlong Shen and Lixin Shen, Orthogonal Fourier-Mellin moments for invariant pattern recognition, J. Opt. Soc. Am. A 11 (6) (1994) p 1748-1757, eq. (6).

T(n, n) = C(2*n-1, n) = A001700(n-1).
T(n, n-1) = A005430(n-1) for n >= 1.
T(n, n-2) = A051133(n-2) for n >= 2.
T(n, 2) = A027480(n-1) for n >= 2.
T(2*n, n) = A208881(n) for n >= 0.
A002002 (row sums).
Cf. A000142, A001286, A001754, A001755, A001777, A182626, A266732 (row k=3), A008316, A033282, A063007, A345013, A269944.

T_row := proc(n) local egf, k, F, t;
if n=0 then RETURN(1) fi;
egf := (x/(1-x))^n/n!; t := diff(egf,[x$n]);
F := convert(t,parfrac,x);
# print(seq(k!*coeff(series(F,x,20),x,k),k=0..7));
# gives A000142, A001286, A001754, A001755, A001777, ...
seq(coeff(F,(x-1)^(-k)),k=n..2*n) end:
seq(print(T_row(n)),n=0..7);
# 2nd version by R. J. Mathar, Dec 18 2016:
A253283 := proc(n,k)
    binomial(n,k)*binomial(n+k-1,k-1) ;
end proc:

Table[Binomial[n, k] Binomial[n + k - 1, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 22 2017 *)

T(n,k) = binomial(n,k)*binomial(n+k-1,k-1);
tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Apr 29 2018

The exponential generating functions for the rows of the square array L(n,k) = ((n+k)!/n!)*C(n+k-1,n-1) (associated to the unsigned Lah numbers) are given by R_n(x) = Sum_{k=0..n} T(n,k)/(x-1)^(n+k).
T(n,k) = C(n,k)*C(n+k-1,k-1).
Sum_{k=0..n} T(n,k) = (-1)^n*hypergeom([-n,n],[1],2) = (-1)^n*A182626(n).
Row generating function: Sum_{k>=1} T(n,k)*z^k = z*n* 2F1(1-n,n+1 ; 2; -z). - R. J. Mathar, Dec 18 2016
From Peter Bala, Feb 22 2017: (Start)
G.f.: (1/2)*( 1 + (1 - t)/sqrt(1 - 2*(2*x + 1)*t + t^2) ) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n,x) = (1/2)*(LegendreP(n, 2*x + 1) - LegendreP(n-1, 2*x + 1)) for n >= 1.
The row polynomials are the black diamond product of the polynomials x^n and x^(n+1) (see Dukes and White 2016 for the definition of this product).
exp(Sum_{n >= 1} R(n,x)*t^n/n) = 1 + x*t + x*(1 + 2*x)*t^2 + x*(1 + 5*x + 5*x^2)*t^3 + ... is a g.f. for A033282, but with a different offset.
The polynomials P(n,x) := (-1)^n/n!*x^(2*n)*(d/dx)^n(1 + 1/x)^n begin 1, 3 + 2*x , 10 + 12*x + 3*x^2, ... and are the row polynomials for the row reverse of this triangle. (End)
Let Q(n, x) = Sum_{j=0..n} (-1)^(n - j)*A269944(n, j)*x^(2*j - 1) and P(x, y) = (LegendreP(x, 2*y + 1) - LegendreP(x-1, 2*y + 1)) / 2 (see Peter Bala above). Then n!*(n - 1)!*[y^n] P(x, y) = Q(n, x) for n >= 1. - Peter Luschny, Oct 31 2022
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 0} binomial(2*n-1, n)*(x*t)^n/(1 - t)^(2*n) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n, x) = [t^n] ( (1 - t)/(1 - (1 + x)*t) )^n.
It follows that for integer x, the sequence {R(n, x) : n >= 0} satisfies the Gauss congruences: R(n*p^r, x) == R(n*p^(r-1), x) (mod p^r) for all primes p and positive integers n and r.
R(n, -2) = (-1)^n * A002003(n) for n >= 1.
R(n, 3) = A299507(n). (End)

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.)

A269947 Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 8, 9, 1, 0, 216, 251, 36, 1, 0, 13824, 16280, 2555, 100, 1, 0, 1728000, 2048824, 335655, 15055, 225, 1, 0, 373248000, 444273984, 74550304, 3587535, 63655, 441, 1, 0, 128024064000, 152759224512, 26015028256, 1305074809, 25421200, 214918, 784, 1

Offset: 0

Peter Luschny, Mar 22 2016

			Triangle starts:
1,
0, 1,
0, 1,       1,
0, 8,       9,       1,
0, 216,     251,     36,     1,
0, 13824,   16280,   2555,   100,   1,
0, 1728000, 2048824, 335655, 15055, 225, 1.

Peter Luschny, The P-transform.

Variant: A249677.
Cf. A007318 (order 0), A132393 (order 1), A269944 (order 2).
Cf. A000442, A000537, A255433.

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + (n-1)^3*T(n-1, k))) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

T[n_, k_] := T[n, k] = Which[n == k, 1, k < 0 || k > n, 0, True, T[n - 1, k - 1] + (n - 1)^3 T[n - 1, k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

T(n,1) = ((n-1)!)^3 for n>=1 (cf. A000442).
T(n,n-1) = (n*(n-1)/2)^2 for n>=1 (cf. A000537).
Row sums: Product_{k=1..n} ((k-1)^3+1) for n>=0 (cf. A255433).

A268434 Triangle read by rows, Lah numbers of order 2, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 10, 10, 1, 0, 100, 140, 28, 1, 0, 1700, 2900, 840, 60, 1, 0, 44200, 85800, 31460, 3300, 110, 1, 0, 1635400, 3476200, 1501500, 203060, 10010, 182, 1, 0, 81770000, 185874000, 90563200, 14700400, 943800, 25480, 280, 1

Offset: 0

Peter Luschny, Mar 07 2016

			[1]
[0,        1]
[0,        2,         1]
[0,       10,        10,        1]
[0,      100,       140,       28,        1]
[0,     1700,      2900,      840,       60,      1]
[0,    44200,     85800,    31460,     3300,    110,     1]
[0,  1635400,   3476200,  1501500,   203060,  10010,   182,   1]

Peter Luschny, The P-transform.

Cf. A038207 (order 0), A111596 (order 1), A269946 (order 3).

Cf. A036969, A105278, A204579, A269938, A269941, A269944, A269945.

T := proc(n,k) option remember;
if n=k then return 1 fi; if k<0 or k>n then return 0 fi;
T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k) end:
seq(seq(T(n,k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A268434_row := n -> PTrans(n, n->`if`(n=1,1, ((n-1)^2+1)/(n*(4*n-2))),
(n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A268434_row(n)), n=0..8);

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^2 + k^2)*T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

#cached_function
def T(n, k):
    if n==k: return 1
    if k<0 or k>n: return 0
    return T(n-1, k-1)+((n-1)^2+k^2)*T(n-1, k)
for n in range(8): print([T(n, k) for k in (0..n)])
# Alternatively with the function PtransMatrix (cf. A269941):
PtransMatrix(8, lambda n: 1 if n==1 else ((n-1)^2+1)/(n*(4*n-2)), lambda n, k: (-1)^k*factorial(2*n)/factorial(2*k))

T(n,k) = (-1)^k*((2*n)!/(2*k)!)*P[n,k](s(n)) where P is the P-transform and s(n) = ((n-1)^2+1)/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n,k) = Sum_{j=k..n} A269944(n,j)*A269945(j,k).
T(n,1) = Product_{k=1..n} (k-1)^2+1 for n>=1 (cf. A101686).
T(n,n-1) = (n-1)*n*(2*n-1)/3 for n>=1 (cf. A006331).
Row sums: A269938.

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

A374848 Obverse convolution A000045**A000045; see Comments.

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A269945 Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

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A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/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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A269947 Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.

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A268434 Triangle read by rows, Lah numbers of order 2, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.

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