A008956 -id:A008956 - cp's OEIS frontend

A160563 Table of the number of (n,k)-Riordan complexes, read by rows.

1, 1, 1, 9, 10, 1, 225, 259, 35, 1, 11025, 12916, 1974, 84, 1, 893025, 1057221, 172810, 8778, 165, 1, 108056025, 128816766, 21967231, 1234948, 28743, 286, 1, 18261468225, 21878089479, 3841278805, 230673443, 6092515, 77077, 455, 1, 4108830350625, 4940831601000

Offset: 0

Jonathan Vos Post, May 19 2009

From Table 4, right-hand side, of Gelineau and Zeng.

Essentially a row-reversal of A008956. - R. J. Mathar, May 20 2009

			Triangle starts:
  [0]         1;
  [1]         1,          1;
  [2]         9,         10,        1;
  [3]       225,        259,       35,        1;
  [4]     11025,      12916,     1974,       84,     1;
  [5]    893025,    1057221,   172810,     8778,   165,    1;
  [6] 108056025,  128816766, 21967231,  1234948, 28743,  286, 1;
.
For row 3: F(x) := 1/cos(x). Then 225*F(x) + 259*(d/dx)^2(F(x)) + 35*(d/dx)^4(F(x)) + (d/dx)^6(F(x)) = 720*(1/cos(x))^7, where F^(r) denotes the r-th derivative of F(x).

Yoann Gelineau and Jiang Zeng, Combinatorial Interpretations of the Jacobi-Stirling Numbers, arXiv:0905.2899 [math.CO], May 2009.
W. Zhang, Some identities involving the Euler and the central factorial numbers, The Fibonacci Quarterly, Vol. 36, Number 2, May 1998.

Cf. A001819, A008275, A008277, A160562, A091885, A182867.

t := proc(n,k) option remember ; expand(x*mul(x+n/2-i,i=1..n-1)) ; coeftayl(%,x=0,k) ; end:
v := proc(n,k) option remember ; 4^(n-k)*t(2*n+1,2*k+1) ; end:
A160563 := proc(n,k) abs(v(n,k)) ; end: for n from 0 to 10 do for k from 0 to n do printf("%d,",A160563(n,k)) ; od: od: # R. J. Mathar, May 20 2009
# Using a bivariate generating function (albeit generating signed terms):
gf := (t + sqrt(1 + t^2))^x: ser := series(gf, t, 20):
ct := n -> coeff(ser, t, n): T := (n, k) -> n!*coeff(ct(n), x, k):
OddPart := (T, len) -> local n, k;
seq(print(seq(T(n, k), k = 1..n, 2)), n = 1..2*len, 2):
OddPart(T, 6);  # Peter Luschny, Mar 03 2024

t[, 0] = 1; t[n, n_] := t[n, n] = ((2*n - 1)!!)^2; t[n_, k_] := t[n, k] = (2*n - 1)^2*t[n - 1, k - 1] + t[n - 1, k];
T[n_, k_] := t[n, n - k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2017, after R. J. Mathar's comment *)

a(n,k) = |v(n,k)| where v(n,k) = v(n-1,k-1) - (2n-1)^2*v(n-1,k); eq (4.2).
Let F(x) = 1/cos(x). Then (2*n)!*(1/cos(x))^(2*n+1) = Sum_{k=0..n} T(n,k)*F^(2*k)(x), where F^(r) denotes the r-th derivative of F(x) (Zhang 1998). An example is given below. - Peter Bala, Feb 06 2012
Given a (0, 0)-based triangle U we call the triangle [U(n, k), k=1..n step 2, n=1..len step 2] the 'odd subtriangle' of U. This triangle is the odd subtriangle of U(n, k) = n! * [x^(n-k)] [t^n] (t + sqrt(1 + t^2))^x, albeit with signed terms. See A182867 for the even subtriangle. - Peter Luschny, Mar 03 2024

Extended by R. J. Mathar, May 20 2009

A162445 A sequence related to the Beta function.

Original entry on oeis.org

1, 8, 384, 46080, 2064384, 3715891200, 392398110720, 1428329123020800, 274239191619993600, 1678343852714360832000, 102043306245033138585600, 4714400748520531002654720000, 160144566965128191597871104000

Offset: 0

Johannes W. Meijer, Jul 06 2009

We define F(z) = Beta(1/2-z/2,1/2+z/2)/Beta(1/2,1/2) = 1/sin(Pi*(1+z)/2) with Beta(z,w) the Beta function. See A008956 for a closely related function.
For the Taylor series expansion of F(z) we can write F(z) = sum(b(n)*(Pi*z)^(2*n)/a(n), n=0..infinity) with b(n) = A046976(n) and a(n) the sequence given above.
We can also write F(z) = sum(c(n)*(Pi*z)^(2*n)/d(n), n=0..infinity) with c(n) = A000364(n) and d(n) = A067624(n).
If p(n) is the exponent of the prime factor 2 in a(n) than p(n) = A120738(n) and 2^p(n) = A061549(n) = abs((4*n)!!/A117972(n)).

Bisection of A050971
Equals 2^(2*n)*A046977(n)
Cf. A008956, A046976, A000364, A067624, A120738, A061549 and A117972.

Denominator[Table[EulerE[2n]/(4n)!!,{n,0,20}]] (* Harvey P. Dale, Jun 23 2013 *)

a(n) = denom(euler(2*n)/(4*n)!!)

A346543 a(n) = [x^n] Product_{k=1..2n} (x + (2k-1)^2).

Original entry on oeis.org

1, 10, 1974, 1234948, 1601489318, 3541644282540, 11934462103156540, 56947950742822581960, 365458809637016986262790, 3035813466162156094097686300, 31694033885101849517370941522644, 406222401519003083851664224927890360, 6271146756206887832796744632163811733084

Offset: 0

Seiichi Manyama, Sep 27 2021

nonn

			(1/3!) * (arcsin(x))^3 = x^3/3! + 10 * x^5/5! + ... . So a(1) =10.
(1/5!) * (arcsin(x))^5 = x^5/5! + 35 * x^7/7! + 1974 * x^9/9! + ... . So a(2) = 1974.

Cf. A008956, A016754, A234324, A293318.

Table[SeriesCoefficient[Product[(x + (2*k-1)^2), {k, 1, 2*n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *)

a(n) = polcoef(prod(k=1, 2*n, x+(2*k-1)^2), n);

a(n) = A008956(2*n,n).
a(n) = (4*n+1)! * [x^(4*n+1)] (1/(2*n+1)!) * (arcsin(x))^(2*n+1).
a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 121.8904568356133798202328777176879971969471503678428704459083316116687149... and c = 0.1081647814943965981694666415038643176470488612855594762896553127... - Vaclav Kotesovec, Oct 16 2021

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);

A380570 Triangle T(n, k) read by rows: Row n gives the coefficients of the even powers in Product_{t=1..n}(2x - (2t - 1))Product_{t=1..n}(2x + (2*t - 1)).

Original entry on oeis.org

1, 4, -1, 16, -40, 9, 64, -560, 1036, -225, 256, -5376, 31584, -51664, 11025, 1024, -42240, 561792, -2764960, 4228884, -893025, 4096, -292864, 7358208, -79036672, 351475696, -515267064, 108056025, 16384, -1863680, 78926848, -1559683840, 14763100352, -61460460880, 87512357916

Offset: 0

Thomas Scheuerle, Jan 27 2025

Odd coefficients of x are excluded here because they are zero.

			Triangle begins:
 n \ k: 0        1        2          3          4           5          6
      x^0      x^2      x^4        x^6        x^8        x^10       x^12
[0]     1;
[1]     4,      -1;
[2]    16,     -40,       9;
[3]    64,    -560,    1036,      -225;
[4]   256,   -5376,   31584,    -51664,     11025;
[5]  1024,  -42240,  561792,  -2764960,   4228884,    -893025;
[6]  4096, -292864, 7358208, -79036672, 351475696, -515267064, 108056025;
     ...

Eric Weisstein's World of Mathematics, Hankel's Symbol.

Cf. A000302 (column 0).
Cf. A001818 (absolute values of main diagonal).
Cf. A001824 (1/4 of absolute values of second diagonal).
Cf. A001825 (1/16 of absolute values of second diagonal).
Cf. A380612 (row sums).
Cf. A008956.

T(n, k) = Vec(prod(k=1,n,2*x-(2*k-1))*prod(k=1,n,2*x+(2*k-1)))[1+2*k]

The Hankel symbol (x, n) is defined as (-1)^n*cos(Pi*x)*Gamma(1/2+n-x)*Gamma(1/2+n+x)/(Pi*n!) = (cos(Pi*x)/((-4)^n*n!))*Sum_{k=0..n} T(n, k)*x^(2*k)..
T(n, k) = A008956(n, k)*4^(n-k)*(-1)^k.
Sum_{k=0..n} T(n, k) = A380612(n) = -(-4)^n*Gamma(-1/2 + n)*Gamma(3/2 + n)/Pi.

cp's OEIS Frontend

A160563 Table of the number of (n,k)-Riordan complexes, read by rows.

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A162445 A sequence related to the Beta function.

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A346543 a(n) = [x^n] Product_{k=1..2*n} (x + (2*k-1)^2).

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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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A380570 Triangle T(n, k) read by rows: Row n gives the coefficients of the even powers in Product_{t=1..n}(2*x - (2*t - 1))*Product_{t=1..n}(2*x + (2*t - 1)).

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A346543 a(n) = [x^n] Product_{k=1..2n} (x + (2k-1)^2).

A380570 Triangle T(n, k) read by rows: Row n gives the coefficients of the even powers in Product_{t=1..n}(2x - (2t - 1))Product_{t=1..n}(2x + (2*t - 1)).