A154602 Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)].

1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 49, 104, 70, 16, 1, 257, 641, 550, 190, 25, 1, 1539, 4380, 4531, 2080, 425, 36, 1, 10299, 32803, 39515, 22491, 6265, 833, 49, 1, 75905, 266768, 365324, 247072, 87206, 16016, 1484, 64, 1, 609441, 2337505, 3575820, 2792476, 1192086, 281190, 36204, 2460, 81, 1

Offset

0,4

Comments

First column is A004211, row sums are A055882.

Triangle T(n,k), read by rows, given by [1,2,1,4,1,6,1,8,1,10,1,12,1,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 02 2009

Links

Table of n, a(n) for n=0..54.

Formula

From Peter Bala, Jun 15 2009: (Start)

T(n,k) = Sum_{i = k..n} 2^(n-i)*binomial(i,k)*Stirling2(n,i).

E.g.f.: exp((t+1)/2*(exp(2*x)-1)) = 1 + (1+t)*x + (3+4*t+t^2)*x^2/2! + ....

Row generating polynomials R_n(x):

R_n(x) = 2^n*Bell(n,(x+1)/2), where Bell(n,x) = Sum_{k = 0..n} Stirling2(n, k)*x^k denotes the n-th Bell polynomial.

Recursion:

R(n+1,x) = (x+1)*(R_n(x) + 2*d/dx(R_n(x))).

(End)

Recurrence: T(n,k) = 2*(k+1)*T(n-1,k+1) + (2*k+1)*T(n-1,k) + T(n-1,k-1). - Emanuele Munarini, Apr 14 2020

Example

Triangle begins
     1;
     1,    1;
     3,    4,    1;
    11,   19,    9,    1;
    49,  104,   70,   16,   1;
   257,  641,  550,  190,  25,  1;
  1539, 4380, 4531, 2080, 425, 36, 1;
Production matrix of this array is
  1,  1,
  2,  3,  1,
  0,  4,  5,  1,
  0,  0,  6,  7,  1,
  0,  0,  0,  8,  9,  1,
  0,  0,  0,  0, 10, 11,  1
with generating function exp(tx)(1+t)(1+2x).

Maple

A154602 := (n, k) -> add(2^(n-j) * binomial(j, k) * Stirling2(n, j), j = k..n): for n from 0 to 6 do seq(A154602(n, k), k = 0..n) od; # Peter Luschny, Dec 13 2022

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[Sinh[#] Exp[#]]&, Sinh[#] Exp[#]&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Crossrefs

Sequence in context: A147721 A172094 A114608 * A216154 A325174 A109956

Adjacent sequences: A154599 A154600 A154601 * A154603 A154604 A154605

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 12 2009

Status

approved