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 A305873 Coefficients of polynomials g_b(x) that arise in the generating function for rooted maps (A053979) 1
 1, 3, 5, 15, 65, 60, 105, 804, 1730, 1105, 945, 10824, 39110, 55645, 27120, 10395, 162357, 854250, 1987270, 2105070, 828250, 135135, 2714445, 19180410, 63897550, 108878610, 91692550, 30220800, 2027025, 50301360, 452984532, 2004435096, 4836052370, 6479714440, 4523710100, 1282031525 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The generating function of the b-th subdiagonal of A053979 is g_b(y)*(1-sqrt(1-4x))/2/(1-4x)^b, b>=0, where g_b(y) = 1 (b= 0 or 1), 3+5*y (b=2), 15+65*y+60*y^2 (b=3) etc are the coefficients in this table, and where y=(1/sqrt(1-4x)-1)/2. The coefficient 794 cited by Walsh-Lehman (1972) has been corrected to 804. LINKS Table of n, a(n) for n=1..36. T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. I, J. Comb. Theory B 13 (1972), 192-218, eq. (5a). MAPLE A305873:= proc(b, x) local gn1, k ; option remember; if b = 0 or b= 1 then return 1 ; else gn1 := procname(b-1, x) ; add(procname(k, x)*procname(b-k, x), k=1..b-1) ; gbx := %*x+(2*(b-1)*(1+2*x)+1)*gn1 ; expand(gbx+2*x*(x+1)*diff(gn1, x)) ; end if; end proc: for b from 1 to 8 do gx := A305873(b, x) ; for l from 0 to b-1 do printf("%d, ", coeff(gx, x, l)) ; end do: printf("\n") ; end do: MATHEMATICA A305873[b_, x_] := A305873[b, x] = Module[{gn1, k, s}, If[b == 0 || b == 1, Return@1, gn1 = A305873[b - 1, x]; s = Sum[A305873[k, x]*A305873[b - k, x], {k, 1, b - 1}]; gbx = s*x + (2*(b - 1)*(1 + 2*x) + 1)*gn1; Expand[gbx + 2*x*(x + 1)*D[gn1, x]]]]; Reap[For[b = 1, b <= 8, b++, gx = A305873[b, x]; For[l = 0, l <= b - 1, l++, Sow[Coefficient[gx, x, l]]]]][[2, 1]] (* Jean-François Alcover, Nov 09 2023, after Maple program *) CROSSREFS Cf. A062980 (diagonal), A001147 (first column) Sequence in context: A018702 A018719 A154107 * A307999 A018771 A214534 Adjacent sequences: A305870 A305871 A305872 * A305874 A305875 A305876 KEYWORD nonn,tabl,easy AUTHOR R. J. Mathar, Jun 12 2018 STATUS approved

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Last modified April 23 09:48 EDT 2024. Contains 371905 sequences. (Running on oeis4.)