A265649 Triangle of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..n} T(n,k)*x^k where T(0,0) = 1, and T(n,k) = 0 for k < 0 or k > n, and T(n,k) = T(n-1,k-1) + (2*n-1+k)*T(n-1,k) for n > 0 and 0 <= k <= n.

1, 1, 1, 3, 5, 1, 15, 33, 12, 1, 105, 279, 141, 22, 1, 945, 2895, 1830, 405, 35, 1, 10395, 35685, 26685, 7500, 930, 51, 1, 135135, 509985, 435960, 146685, 23310, 1848, 70, 1, 2027025, 8294895, 7921305, 3076290, 589575, 60270, 3318, 92, 1, 34459425, 151335135, 158799690, 69447105, 15457365, 1915515, 136584, 5526, 117, 1

Offset: 0

Author

Werner Schulte, Dec 11 2015

Keywords

• nonn
• easy
• tabl

Comments

The polynomials p(n,x) satisfy the differential equation: x*y''' + (3*x+1)*y'' + (2*x+2)*y' - 2*n*y = 0 where y' = dy/dx (first derivative).

Appears to be the exponential Riordan array [1/sqrt(1 - 2x), 1/(sqrt(1 - 2x) - 1)]. [Barry, Example 1] - Eric M. Schmidt, Sep 23 2017

Examples

			The triangle T(n,k) begins:
n\k:        0        1        2        3       4      5     6   7  8
  0:        1
  1:        1        1
  2:        3        5        1
  3:       15       33       12        1
  4:      105      279      141       22       1
  5:      945     2895     1830      405      35      1
  6:    10395    35685    26685     7500     930     51     1
  7:   135135   509985   435960   146685   23310   1848    70   1
  8:  2027025  8294895  7921305  3076290  589575  60270  3318  92  1
  etc.
The polynomial corresponding to row 3 is p(3,x) = 15 + 33*x + 12*x^2 + x^3.
		

Links

• Paul Barry, A Note on d-Hankel Transforms, Continued Fractions, and Riordan Arrays, arXiv:1702.04011 [math.CO], 2017.
• Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See pp. 21, 28.

Crossrefs

Cf. A000007, A000326, A001147, A001497, A021009, A129890, A035342.

Programs

Maple

T := (n, k) -> local j; 2^n*add((-1)^(k-j)*binomial(k, j)*pochhammer((j+1)/2, n), j=0..k) / k!: for n from 0 to 6 do seq(T(n, k), k=0..n) od;  # Peter Luschny, Mar 04 2024

Mathematica

(* The function RiordanArray is defined in A256893. *)
rows = 10;
R = RiordanArray[1/Sqrt[1 - 2 #]&, 1/Sqrt[1 - 2 #] - 1&, rows, True];
R // Flatten (* Jean-François Alcover, Jul 20 2019 *)

Formula

Recurrence: p(0,x) = 1 and p(n+1,x) = (2*n+1+x)*p(n,x) + x*p'(n,x).

T(n,0) = A001147(n), T(n+1,1) = A129890(n), T(n+1,n) = A000326(n+1), and Sum_{k=0..n} (-1)^k*k!*T(n,k) = A000007(n).

Recurrence: k^2*(k+1)*T(n,k+1) = (2*n+2-2*k)*T(n,k-1)-k*(3*k-1)*T(n,k).

Conjecture: T(n,k) = 2^(n-k)*(n-k)!*binomial(n,k)*(Sum_{j=0..n-k} (-1/4)^j* binomial(2*j+k,j)*binomial(n,j+k)).

Conjecture: T(n,k) = (-1)^k*Sum_{j=0..n-1} A001497(n-1,j)*A021009(j+1,k).

T(n,k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i)*Product_{j=1..n} (2*j+i-1))/k!. - Werner Schulte, Mar 03 2024

T(n,k) = (2^n/k!)*(Sum_{j=0..k}(-1)^(k-j)*binomial(k,j)*Pochhammer((j + 1)/2, n)). - Peter Luschny, Mar 04 2024