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

%I A154602 #35 Apr 19 2025 11:06:41
%S A154602 1,1,1,3,4,1,11,19,9,1,49,104,70,16,1,257,641,550,190,25,1,1539,4380,
%T A154602 4531,2080,425,36,1,10299,32803,39515,22491,6265,833,49,1,75905,
%U A154602 266768,365324,247072,87206,16016,1484,64,1,609441,2337505,3575820,2792476,1192086,281190,36204,2460,81,1
%N A154602 Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)].
%C A154602 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
%H A154602 G. C. Greubel, <a href="/A154602/b154602.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A154602 T(n, 0) = A004211(n).
%F A154602 Sum_{k=0..n} T(n, k) = A055882(n) (row sums).
%F A154602 From _Peter Bala_, Jun 15 2009: (Start)
%F A154602 T(n,k) = Sum_{i = k..n} 2^(n-i)*binomial(i,k)*Stirling2(n,i).
%F A154602 E.g.f.: exp((t+1)/2*(exp(2*x)-1)) = 1 + (1+t)*x + (3+4*t+t^2)*x^2/2! + ....
%F A154602 Row generating polynomials R_n(x):
%F A154602 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.
%F A154602 Recursion:
%F A154602 R(n+1,x) = (x+1)*(R_n(x) + 2*d/dx(R_n(x))).
%F A154602 (End)
%F A154602 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
%F A154602 Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n). - _G. C. Greubel_, Sep 19 2024
%F A154602 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
%e A154602 Triangle begins
%e A154602      1;
%e A154602      1,    1;
%e A154602      3,    4,    1;
%e A154602     11,   19,    9,    1;
%e A154602     49,  104,   70,   16,   1;
%e A154602    257,  641,  550,  190,  25,  1;
%e A154602   1539, 4380, 4531, 2080, 425, 36, 1;
%e A154602 Production matrix of this array is
%e A154602   1,  1,
%e A154602   2,  3,  1,
%e A154602   0,  4,  5,  1,
%e A154602   0,  0,  6,  7,  1,
%e A154602   0,  0,  0,  8,  9,  1,
%e A154602   0,  0,  0,  0, 10, 11,  1
%e A154602 with generating function exp(t*x)*(1+t)*(1+2*x).
%p A154602 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
%t A154602 (* The function RiordanArray is defined in A256893. *)
%t A154602 RiordanArray[Exp[Sinh[#] Exp[#]]&, Sinh[#] Exp[#]&, 10, True] // Flatten (* _Jean-François Alcover_, Jul 19 2019 *)
%o A154602 (Magma)
%o A154602 A154602:= func< n,k | (&+[2^(n-j)*Binomial(j,k)*StirlingSecond(n,j): j in [k..n]]) >;
%o A154602 [A154602(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 19 2024
%o A154602 (SageMath)
%o A154602 def A154602(n,k): return sum(2^(n-j)*binomial(j,k)* stirling_number2(n,j) for j in range(k,n+1))
%o A154602 flatten([[A154602(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 19 2024
%Y A154602 Columns k=0..3 give A000007, A383203, A383204, A383205.
%Y A154602 Cf. A004211 (first column), A256893.
%Y A154602 Sums include: A000007 (alternating sign row), A055882 (row sums).
%K A154602 easy,nonn,tabl
%O A154602 0,4
%A A154602 _Paul Barry_, Jan 12 2009