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A111578 Triangle T(n, m) = T(n-1, m-1) + (4m-3)*T(n-1, m) read by rows 1<=m<=n. 7
1, 1, 1, 1, 6, 1, 1, 31, 15, 1, 1, 156, 166, 28, 1, 1, 781, 1650, 530, 45, 1, 1, 3906, 15631, 8540, 1295, 66, 1, 1, 19531, 144585, 126651, 30555, 2681, 91, 1, 1, 97656, 1320796, 1791048, 646086, 86856, 4956, 120, 1, 1, 488281, 11984820, 24604420, 12774510 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
From Peter Bala, Jan 27 2015: (Start)
Working with an offset of 0, this is the exponential Riordan array [exp(z), (exp(4*z) - 1)/4].
This is the triangle of connection constants between the polynomial basis sequences {x^n}n>=0 and { n!*4^n * binomial((x - 1)/4,n) }n>=0. An example is given below.
Call this array M and let P denote Pascal's triangle A007318 then P^2 * M = A225469; P^(-1) * M is a shifted version of A075499.
This triangle is the particular case a = 4, b = 0, c = 1 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. (End)
LINKS
FORMULA
From Peter Bala, Jan 27 2015: (Start)
The following formulas assume an offset of 0.
T(n,k) = 1/(4^k*k!)*sum {j = 0..k} (-1)^(k-j)*binomial(k,j)*(4*j + 1)^n.
T(n,k) = sum {i = 0..n-1} 4^(i-k+1)*binomial(n-1,i)*Stirling2(i,k-1).
E.g.f.: exp(z)*exp(x/4*(exp(4*z) - 1)) = 1 + (1 + x)*z + (1 + 6*x + x^2)*z^2/2! + ....
O.g.f. for n-th diagonal: exp(-x/4)*sum {k >= 0} (4*k + 1)^(k + n - 1)*((x/4*exp(-x))^k)/k!.
O.g.f. column k: 1/( (1 - x)*(1 - 5*x)...(1 - (4*k + 1)*x ). (End)
EXAMPLE
The triangle starts in row n=1 as:
1;
1,1;
1,6,1;
1,31,15,1;
Connection constants: Row 4: [1, 31, 15, 1] so
x^3 = 1 + 31*(x - 1) + 15*(x - 1)*(x - 5) + (x - 1)*(x - 5)*(x - 9). - Peter Bala, Jan 27 2015
MATHEMATICA
T[n_, k_] := 1/(4^(k-1)*(k-1)!) * Sum[ (-1)^(k-j-1) * (4*j+1)^(n-1) * Binomial[k-1, j], {j, 0, k-1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Peter Bala *)
PROG
(Python)
def A096038(n, m):
if n < 1 or m < 1 or m > n:
return 0
elif n <=2:
return 1
else:
return A096038(n-1, m-1)+(4*m-3)*A096038(n-1, m)
print( [A096038(n, m) for n in range(20) for m in range(1, n+1)] )
# R. J. Mathar, Oct 11 2009
CROSSREFS
Cf. A111577, A008277, A039755, A016234 (3rd column).
Sequence in context: A368848 A265603 A174186 * A166349 A176429 A157155
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Aug 07 2005
EXTENSIONS
Edited and extended by R. J. Mathar, Oct 11 2009
STATUS
approved

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Last modified June 25 13:13 EDT 2024. Contains 373705 sequences. (Running on oeis4.)