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A154602
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Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)]
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0
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 02 2009]
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FORMULA
| Contribution from Peter Bala (pbala(AT)talktalk.net), Jun 15 2009: (Start)
TABLE ENTRIES
T(n,k) = sum {i = k..n} 2^(n-i)*binomial(i,k)*Stirling2(n,i).
GENERATING FUNCTIONS
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)
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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).
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CROSSREFS
| Sequence in context: A147721 A172094 A114608 * A109956 A123319 A076785
Adjacent sequences: A154599 A154600 A154601 * A154603 A154604 A154605
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Jan 12 2009
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