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A152650
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Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of certain polynomials P(n,x).
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5
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1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 9, 4, 1, 1, 2, 9, 8, 5, 1, 1, 4, 27, 32, 25, 6, 1, 1, 4, 81, 32, 125, 18, 7, 1, 1, 8, 81, 128, 625, 36, 49, 8, 1, 1, 2, 243, 256, 625, 54, 343, 32, 9, 1, 1, 4, 729, 1024, 3125, 324, 2401, 256, 81, 10, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Let the polynomials P be defined by P(0,x)=u(0), P(n,x)= u(n) + x*sum_{i=0..n-1} u(i)*P(n-i-1,x)
and coefficients u(i)=1/i!. These u are reminiscent of the
Taylor expansion of exp(x). Then P(n,x) = sum_{k=0..n} c(n,k)*x^k.
n!*P(n,x) are the row polynomials of A152818. - Peter Bala, Oct 09 2011
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EXAMPLE
| The triangle c(n,k) and polynomials start in row n = 0 as:
1 = 1 ;
1, 1, = 1+x
1/2, 2, 1, = 1/2+2*x+x^2
1/6, 2, 3, 1, = 1/6+2*x+3*x^2+x^3
1/24, 4/3, 9/2, 4, 1, = 1/24+4/3*x+9/2*x^2+4*x^3+x^4
1/120, 2/3, 9/2, 8, 5, 1, = 1/120+2/3*x+9/2*x^2+8*x^3+5*x^4+x^5
1/720, 4/15, 27/8, 32/3, 25/2, 6, 1, = 1/720+4/15*x+27/8*x^2+32/3*x^3+25/2*x^4+6*x^5+x^6
1/5040, 4/45, 81/40, 32/3, 125/6, 18, 7, 1 = 1/5040+4/45*x+32/3*x^3+81/40*x^2+125/6*x^4+18*x^5+7*x^6+x^7
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MAPLE
| u := proc(i) 1/i! end:
P := proc(n, x) option remember ; if n =0 then u(0); else u(n)+x*add( u(i)*procname(n-1-i, x), i=0..n-1) ; expand(%) ; fi; end:
A152650 := proc(n, k) p := P(n, x) ; numer(coeftayl(p, x=0, k)) ; end:
seq(seq(A152650(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Aug 24 2009
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CROSSREFS
| Cf. A152656 (denominators), A140749, A141412, A141904, A142048. A152818.
Sequence in context: A156041 A133255 A145972 * A184219 A180262 A161789
Adjacent sequences: A152647 A152648 A152649 * A152651 A152652 A152653
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KEYWORD
| nonn,frac,tabl
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AUTHOR
| Paul Curtz (bpcrtz(AT)free.fr), Dec 10 2008
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EXTENSIONS
| Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 24 2009
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