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A134991
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Triangle of Ward numbers T(n,k) read by rows.
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18
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1, 1, 3, 1, 10, 15, 1, 25, 105, 105, 1, 56, 490, 1260, 945, 1, 119, 1918, 9450, 17325, 10395, 1, 246, 6825, 56980, 190575, 270270, 135135, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025, 1, 1012, 74316, 1487200, 12122110, 47507460
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OFFSET
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1,3
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COMMENTS
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This is a reordered version of A008299 read along the diagonals (see table on p. 222 in Comtet) and a row-reversed version of a table on p. 92 in the Ward reference (and A181996, with an additional leading 1). A134685 is a refinement of the Ward table. The first and second diagonals are A001147 and A000457 and appear in the diagonals of several OEIS entries. The polynomials also appear in Carlitz (p. 85), Drake et al. (p. 8) and Smiley (p. 7).
First few polynomials (with a different offset) are
P(0,t) = 0
P(1,t) = 1
P(2,t) = t
P(3,t) = t + 3 t^2
P(4,t) = t + 10 t^2 + 15 t^3
P(5,t) = t + 25 t^2 + 105 t^3 + 105 t^4
These are the "face" numbers of the tropical Grassmannian G(2,n),related to phylogenetic trees (with offset 0 beginning with P(2,t)). Corresponding h-vectors are A008517. - Tom Copeland, Oct 03 2011
A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n , for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=-t, and (a_n)=-(1+t) P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 =0. - Tom Copeland, Oct 08 2011
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REFERENCES
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L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., vol 35 (1968), p. 83-90.
Clark, Lane, Asymptotic normality of the Ward numbers. Discrete Math. 203 (1999), no. 1-3, 41-48. [From N. J. A. Sloane, Feb 06 2012]
L. Comtet, Advanced Combinatorics, Reidel, 1974.
B. Drake, I. M. Gessel and G. Xin, Three proofs and a generalization of the Goulden-Litsyn-Shevelev conjecture ..., J. Integer Sequences, Vol. 10 (2007), #07.3.7.
M. Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., 56 (1934), p. 87-95.
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LINKS
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Table of n, a(n) for n=1..42.
MathOverflow, Face numbers for tropical Grassmannian G'_2,7 simplicial complex?
L. M. Smiley, Completion of a Rational Function Sequence of Carlitz
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FORMULA
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E.g.f. for the polynomials is A(x,t) = (x-t)/(t+1) + T{ (t/(t+1)) * exp[(x-t)/(t+1)] }, where T(x) is the Tree function, the e.g.f. of A000169. The compositional inverse in x (about x = 0) is B(x) = x + -t * [exp(x) - x - 1]. Special case t = 1 gives e.g.f. for A000311. These results are a special case of A134685 with u(x) = B(x).
Contribution from Tom Copeland, Oct 26 2008: (Start)
Umbral-Sheffer formalism gives, for m a positive integer and u = t/(t+1),
[P(.,t)+Q(.,x)]^m = [m Q(m-1,x) - t Q(m,x)]/(t+1) + sum(n>=1) { n^(n-1)[u exp(-u)]^n/n! [n/(t+1)+Q(.,x)]^m }, when the series is convergent for a sequence of functions Q(n,x).
Check: With t=1; Q(n,x)=0^n, for n>=0; and Q(-1,x)=0, then [P(.,1)+Q(.,x)]^m = P(m,1) = A000311(m).
(End)
Let h(x,t) = 1/(dB(x)/dx) = 1/(1-t*(exp(x)-1)), an e.g.f. in x for row polynomials in t of A019538, then the n-th row polynomial in t of the table A134991, P(n,t), is given by ((h(x,t)*d/dx)^n)x evaluated at x=0, i.e., A(x,t) = exp(x*P(.,t)) = exp(x*h(u,t)*d/du) u evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t). - Tom Copeland, Sep 05 2011
The polynomials (1+t)/t*P(n,t) are the row polynomials of A112493. Let f(x) = (1+x)/(1-x*t). Then for n >= 0, P(n+1,t) is given by t/(1+t)*(f(x)*d/dx)^n(f(x)) evaluated at x = 0. - Peter Bala, Sep 30 2011
Contributions from Tom Copeland, Oct 04 2011: (Start)
a(n,k)=(k+1)a(n-1,k)+(n+k+1)a(n-1,k-1) with starting indices n=0 and k=0 beginning with P(2,t) (as suggested by a formula of David Speyer on MathOverflow). a(n,k)= k a(n-1,k)+(n+k-1)a(n-1,k-1) with starting indices n=1 and k=1 of table (cf. Smiley above and Riordin ref.[10] therein).
P(n,t) = (1/(1+t))^n sum(k=1 to infin) k^(n+k-1) [(u*exp(-u)]^k / k! with u=(t/(t+1)) for n>1; therefore, sum(k=1 to infin) (-1)^k k^(n+k-1) x^k/k!
= [1+LW(x)]^(-n) P{n,-LW(x)/[1+LW(x)]}, with LW(x) the Lambert W-Fct.
a(n,k)= sum(i=0 to k) {(-1)^i binom(n+k,i) sum(j=0 to k-i) (-1)^j (k-i-j)^(n+k-i)/[j!(k-i-j)!]} from relation to A008299. (End)
The e.g.f. A(x,t) = -v * { sum(j=1 to infin) D(j-1,u) (-z)^j / j! } where u=(x-t)/(1+t) , v=1+u, z= x/[(1+t) v^2] and D(j-1,u) are the polynomials of A042977. dA/dx= 1/[(1+t)(v-A)]= 1/{1-t*[exp(A)-1]}.- Tom Copeland, Oct 06 2011
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CROSSREFS
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Cf. A112493. A181996 is the mirror image.
Sequence in context: A068438 A064060 A176740 * A212930 A225725 A095327
Adjacent sequences: A134988 A134989 A134990 * A134992 A134993 A134994
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KEYWORD
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nonn,tabl
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AUTHOR
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Tom Copeland, Feb 05 2008
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EXTENSIONS
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Edited by N. J. A. Sloane, Apr 05 2012
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STATUS
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approved
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