A204579 -id:A204579 - cp's OEIS frontend

A036969 Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.

1, 1, 1, 1, 5, 1, 1, 21, 14, 1, 1, 85, 147, 30, 1, 1, 341, 1408, 627, 55, 1, 1, 1365, 13013, 11440, 2002, 91, 1, 1, 5461, 118482, 196053, 61490, 5278, 140, 1, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1, 1, 87381, 9668036, 53157079, 46587905

Offset: 1

N. J. A. Sloane

Or, triangle of central factorial numbers T(2n,2k) (in Riordan's notation).
Can be used to calculate the Bernoulli numbers via the formula B_2n = (1/2)*Sum_{k = 1..n} (-1)^(k+1)*(k-1)!*k!*T(n,k)/(2*k+1). E.g., n = 1: B_2 = (1/2)*1/3 = 1/6. n = 2: B_4 = (1/2)*(1/3 - 2/5) = -1/30. n = 3: B_6 = (1/2)*(1/3 - 2*5/5 + 2*6/7) = 1/42. - Philippe Deléham, Nov 13 2003
From Peter Bala, Sep 27 2012: (Start)
Generalized Stirling numbers of the second kind. T(n,k) is equal to the number of partitions of the set {1,1',2,2',...,n,n'} into k disjoint nonempty subsets V1,...,Vk such that, for each 1 <= j <= k, if i is the least integer such that either i or i' belongs to Vj then {i,i'} is a subset of Vj. An example is given below.
Thus T(n,k) may be thought of as a two-colored Stirling number of the second kind. See Matsumoto and Novak, who also give another combinatorial interpretation of these numbers. (End)

			Triangle begins:
  1;
  1,    1;
  1,    5,      1;
  1,   21,     14,      1;
  1,   85,    147,     30,     1;
  1,  341,   1408,    627,    55,    1;
  1, 1365,  13013,  11440,  2002,   91,   1;
  1, 5461, 118482, 196053, 61490, 5278, 140, 1;
  ...
T(3,2) = 5: The five set partitions into two sets are {1,1',2,2'}{3,3'}, {1,1',3,3'}{2,2'}, {1,1'}{2,2',3,3'}, {1,1',3}{2,2',3'} and {1,1',3'}{2,2',3}.

L. Carlitz, A conjecture concerning Genocchi numbers. Norske Vid. Selsk. Skr. (Trondheim) 1971, no. 9, 4 pp. [The triangle appears on page 2.]
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.

Vincenzo Librandi, Rows n = 1..100 of triangle, flattened
Thomas Browning, Counting Parabolic Double Cosets in Symmetric Groups, arXiv:2010.13256 [math.CO], 2020.
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.
José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 2.
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.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)
Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, Alternating runs of permutations and the central factorial numbers, arXiv:2202.13978 [math.CO], 2022.
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265. - From _N. J. A. Sloane_, Jan 02 2013
Petro Kolosov, Polynomial identities involving central factorial numbers, GitHub, 2024. See p. 6.
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.
S. Matsumoto and J. Novak, Jucys-Murphy Elements and Unitary Matrix Integrals arXiv.0905.1992 [math.CO], 2009-2012.
B. K. Miceli, Two q-Analogues of Poly-Stirling Numbers, J. Integer Seq., 14 (2011), 11.9.6.
John Riordan, Letter, Apr 28 1976.
John Riordan, Letter, Jul 06 1978
Richard P. Stanley, Hook Lengths and Contents.

Columns are A002450, A002451.
Diagonals are A000330 and A060493.
Transpose of A008957.
(0,0)-based version: A269945.
Cf. A008955, A008956, A156289, A135920 (row sums), A204579 (inverse), A000290.

a036969 n k = a036969_tabl !! (n-1) (k-1)
a036969_row n = a036969_tabl !! (n-1)
a036969_tabl = iterate f [1] where
   f row = zipWith (+)
     ([0] ++ row) (zipWith (*) (tail a000290_list) (row ++ [0]))
-- Reinhard Zumkeller, Feb 18 2013

A036969 := proc(n,k) local j; 2*add(j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!),j=1..k); end;

t[n_, k_] := 2*Sum[j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!), {j, 1, k}]; Flatten[ Table[t[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Oct 11 2011 *)
t1[n_, k_] := (1/(2 k)!) * Sum[Binomial[2 k, j]*(-1)^j*(k - j)^(2 n), {j, 0, 2 k}]; Column[Table[t1[n, k], {n, 1, 10}, {k, 1, n}]] (* Kolosov Petro ,Jul 26 2023 *)

PARI
```
T(n,k)=if(1M. F. Hasler, Feb 03 2012
```

T(n,k)=2*sum(j=1,k,(-1)^(k-j)*j^(2*n)/(k-j)!/(k+j)!)  \\ M. F. Hasler, Feb 03 2012

def A036969(n,k) : return (2/factorial(2*k))*add((-1)^j*binomial(2*k,j)*(k-j)^(2*n) for j in (0..k))
for n in (1..7) : print([A036969(n,k) for k in (1..n)]) # Peter Luschny, Feb 03 2012

T(n,k) = A156289(n,k)/A001147(k). - Peter Bala, Feb 21 2011
From Peter Bala, Oct 14 2011: (Start)
O.g.f.: Sum_{n >= 1} x^n*t^n/Product_{k = 1..n} (1 - k^2*t^2) = x*t + (x + x^2)*t^2 + (x + 5*x^2 + x^3)*t^3 + ....
Define polynomials x^[2*n] = Product_{k = 0..n-1} (x^2 - k^2). This triangle gives the coefficients in the expansion of the monomials x^(2*n) as a linear combination of x^[2*m], 1 <= m <= n. For example, row 4 gives x^8 = x^[2] + 21*x^[4] + 14*x^[6] + x^[8].
A008955 is a signed version of the inverse.
The n-th row sum = A135920(n). (End)
T(n,k) = (2/(2*k)!)*Sum_{j=0..k-1} (-1)^(j+k+1) * binomial(2*k,j+k+1) * (j+1)^(2*n). This formula is valid for n >= 0 and 0 <= k <= n. - Peter Luschny, Feb 03 2012
From Peter Bala, Sep 27 2012: (Start)
Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/((2*n)!/2^n). A generating function for the triangle is E(t*(E(x)-1)) = 1 + t*x + t*(1 + t)*x^2/6 + t*(1 + 5*t + t^2)*x^3/90 + ..., where the sequence of denominators [1, 1, 6, 90, ...] is given by (2*n)!/2^n. Cf. A008277 which has generating function exp(t*(exp(x)-1)). An e.g.f. is E(t*(E(x^2/2)-1)) = 1 + t*x^2/2! + t*(1 + t)*x^4/4! + t*(1 + 5*t + t^2)*x^6/6! + ....
Put c(n) := (2*n)!/2^n. The column k generating function is (1/c(k))*(E(x)-1)^k = Sum_{n >= k} T(n,k)*x^n/c(n). The inverse array is A204579.
The production array begins:
  1,  1;
  0,  4,  1;
  0,  0,  9,  1;
  0,  0,  0, 16,  1;
  ... (End)
x^n = Sum_{k=1..n} T(n,k)*Product_{i=0..k-1} (x-i^2), see Stanley link. - Michel Marcus, Nov 19 2014; corrected by Kolosov Petro, Jul 26 2023
From Kolosov Petro, Jul 26 2023: (Start)
T(n,k) = (1/(2*k)!) * Sum_{j=0..2k} binomial(2k, j)*(-1)^j*(k - j)^(2n).
T(n,k) = (1/(k*(2k-1)!)) * Sum_{j=0..k} (-1)^(k-j)*binomial(2k, k-j)*j^(2n). (End)

More terms from Vladeta Jovovic, Apr 16 2000

A269944 Triangle read by rows, Stirling cycle 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*T(n-1, k), for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 5, 1, 0, 36, 49, 14, 1, 0, 576, 820, 273, 30, 1, 0, 14400, 21076, 7645, 1023, 55, 1, 0, 518400, 773136, 296296, 44473, 3003, 91, 1, 0, 25401600, 38402064, 15291640, 2475473, 191620, 7462, 140, 1

Offset: 0

Peter Luschny, Mar 22 2016

Also known as central factorial numbers |t(2*n, 2*k)| (cf. A008955).

The analog for the Stirling set numbers is A269945.

			Triangle starts:
  [1]
  [0,     1]
  [0,     1,     1]
  [0,     4,     5,    1]
  [0,    36,    49,   14,    1]
  [0,   576,   820,  273,   30,  1]
  [0, 14400, 21076, 7645, 1023, 55, 1]

Peter Luschny, The P-transform.
Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.
B. K. Miceli, Two q-Analogues of Poly-Stirling Numbers, J. Integer Seq., 14 (2011), 11.9.6.

Variants: A204579 (signed, row 0 missing), A008955.
Cf. A007318 (order 0), A132393 (order 1), A269947 (order 3).
Cf. A000330 (subdiagonal), A001044 (column 1), A101686 (row sums), A269945 (Stirling set), A269941 (P-transform).

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*T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A269944_row := n -> PTrans(n, n->`if`(n=1, 1, (n-1)^2/(n*(4*n-2))), (n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269944_row(n)), n=0..8);
# From Peter Luschny, Feb 29 2024: (Start)
# Computed as the coefficients of polynomials:
P := (x, n) -> local j; mul((j - x)*(j + x), j = 0..n-1):
T := (n, k) -> (-1)^k*coeff(P(x, n), x, 2*k):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
# Alternative, using the exponential generating function:
egf := cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)):
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, n-k), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # (End)
# Alternative, row polynomials:
rowpoly := n -> pochhammer(-sqrt(x), n) * pochhammer(sqrt(x), n):
row := n -> local k; seq((-1)^k*coeff(expand(rowpoly(n)), x, k), k = 0..n):
seq(print(row(n)), n = 0..6);  # Peter Luschny, Aug 03 2024

T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + (n - 1)^2*T[n - 1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Jul 25 2019 *)

stircycle2 = lambda n: 1 if n == 1 else (n-1)^2/(n*(4*n-2))
norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stircycle2, 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) = (n - 1)^2 / (n*(4*n - 2)). The P-transform is defined in the link. See the Sage and Maple implementations below.
T(n, 1) = ((n - 1)!)^2 for n >= 1 (cf. A001044).
T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
Row sums: Product_{k=1..n} ((k - 1)^2 + 1) for n >= 0 (cf. A101686).
From Fabián Pereyra, Apr 25 2022: (Start)
T(n,k) = (-1)^(n-k)*Sum_{j=2*k..2*n} Stirling1(2*n,j)*binomial(j,2*k)*(n-1)^(j-2*k).
T(n,k) = Sum_{j=0..2*k} (-1)^(j - k)*Stirling1(n, j)*Stirling1(n, 2*k - j). (End)
From Peter Luschny, Feb 29 2024: (Start)
T(n, k) = (-1)^k*[x^(2*k)] P(x, n) where P(x, n) = Product_{j=0..n-1} (j-x)*(j+x).
T(n, k) = (2*n)!*[t^(n-k)] [x^(2*n)] cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)). (End)
T(n, k) = (-1)^k*[x^k] Pochhammer(-sqrt(x), n) * Pochhammer(sqrt(x), n). - Peter Luschny, Aug 03 2024

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

A036969 Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.

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A269944 Triangle read by rows, Stirling cycle 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*T(n-1, k), for 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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