OFFSET
0,4
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
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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
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(t*x)*(1+t)*(1+2*x).
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jan 12 2009
STATUS
approved