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

Author

Paul Barry, Jan 12 2009

Keywords

• easy
• nonn
• tabl

Comments

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

Examples

			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(t*x)*(1+t)*(1+2*x).
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Columns k=0..3 give A000007, A383203, A383204, A383205.

Cf. A004211 (first column), A256893.

Sums include: A000007 (alternating sign row), A055882 (row sums).

Programs

Magma

A154602:= func< n,k | (&+[2^(n-j)*Binomial(j,k)*StirlingSecond(n,j): j in [k..n]]) >;
[A154602(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 19 2024

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 *)

SageMath

def A154602(n,k): return sum(2^(n-j)*binomial(j,k)* stirling_number2(n,j) for j in range(k,n+1))
flatten([[A154602(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 19 2024

Formula

T(n, 0) = A004211(n).

Sum_{k=0..n} T(n, k) = A055882(n) (row sums).

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

Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n). - G. C. Greubel, Sep 19 2024

E.g.f. of column k (with leading zeros): f(x)^k * exp(f(x)) / k! with f(x) = (exp(2*x) - 1)/2. - Seiichi Manyama, Apr 19 2025