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A135494
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Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with lowering operator (D-1)/2 + T{ (1/2) * exp[(D-1)/2] } where T(x) is Cayley's Tree function.
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2
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1, -1, 1, -1, -3, 1, -1, -1, -6, 1, -1, 5, 5, -10, 1, -1, 19, 30, 25, -15, 1, -1, 49, 49, 70, 70, -21, 1, -1, 111, -70, -91, 70, 154, -28, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| The lowering (or delta) operator for these polynomials is L = (D-1)/2 + T{ (1/2) * exp[(D-1)/2] } and the raising operator is R = 2t * { 1 - T[ (1/2) * exp[(D-1)/2] ] }, where T(x) is the tree function of A000169. In addition, L = E(D,1) = A(D) where E(x,t) is the e.g.f. of A134991 and A(x) is the e.g.f. of A000311, so L = sum(j=1,...) A000311(j) * D^j / j! also. The polynomials and operators can be generalized through A134991.
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REFERENCES
| S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.
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FORMULA
| Row polynomials are P(n,t) = sum(j=1,...,n) C(n,j) * x^j = [ Bell(.,-t) + 2t ]^n,umbrally, where Bell(j,t) are the Touchard/Bell/exponential polynomials described in A008277, with P(0,t) = 1 .
The e.g.f. is exp{ t * [ -exp(x) + 2x + 1] } and [ P(.,t) + P(.,s) ]^n = P(n,s+t) .
The lowering operator gives L[P(n,t)] = n * P(n-1,t) = (D-1)/2 * P(n,t) + sum(j=1,...) j^(j-1) * 2^(-j) / j! * exp(-j/2) * P(n,t + j/2) .
The raising operator gives R[P(n,t)] = P(n+1,t) = 2t * { P(n,t) - sum(j=1,...) j^(j-1) * 2^(-j) / j! * exp(-j/2) * P(n,t + j/2) } .
Therefore P(n+1,t) = 2t * { [ (1+D)/2 * P(n,t) ] - n * P(n-1,t) } .
P(n,1) = (-1)^n * A074051(n) and P(n,-1) = A126617(n) .
See Rota, Roman, Mathworld or Wikipedia on Sheffer sequences and umbral calculus for more formulae, including expansion theorems.
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CROSSREFS
| Sequence in context: A046643 A112475 A126799 * A016566 A096744 A180051
Adjacent sequences: A135491 A135492 A135493 * A135495 A135496 A135497
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KEYWORD
| sign,tabl
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AUTHOR
| Tom Copeland (tcjpn(AT)msn.com), Feb 08 2008
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