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A191685 Eighth diagonal a(n) = s(n,n-7) of the unsigned Stirling numbers of the first kind with n>7.

%I A191685 #19 Jun 02 2025 04:10:22
%S A191685 5040,109584,1172700,8409500,45995730,206070150,790943153,2681453775,
%T A191685 8207628000,23057159840,60202693980,147560703732,342252511900,
%U A191685 756111184500,1599718388730,3256091103430,6400590336096,12191224980000,22563937825000,40681506808800
%N A191685 Eighth diagonal a(n) = s(n,n-7) of the unsigned Stirling numbers of the first kind with n>7.
%C A191685 The Maple programs under I generate the sequence. The Maple program under II generates explicit formulas for a(n+1) = s(n+1,n+1-c) with c>=1 and n>=c.
%D A191685 K. Seidel, Variation der Binomialkoeffizienten, Bild
%D A191685 der Wissenschaft, 6 (1980), pp. 127-128.
%H A191685 T. D. Noe, <a href="/A191685/b191685.txt">Table of n, a(n) for n = 8..1000</a>
%F A191685 a(n+1) = A130534(T(n,n-7)) = s(n+1,n+1-7)
%F A191685 a(n+1) = binomial(n+1,8)*(80*n+114*n^2-23*n^3-75*n^4-9*n^5+9*n^6)/144
%e A191685 c=1; a(n+1) = binomial(n+1,2)
%e A191685 c=2; a(n+1) = binomial(n+1,3)*(2+3*n)/4
%e A191685 c=3; a(n+1) = binomial(n+1,4)*(n+n^2)/2
%e A191685 c=4; a(n+1) = binomial(n+1,5)*(-8-10*n+15*n^2 +15*n^3)/48
%e A191685 c=5; a(n+1) = binomial(n+1,6)*(-6*n-7*n^2+2*n^3+ 3*n^4)/16
%e A191685 c=6; a(n+1) = binomial(n+1,7)*(96+140*n-224*n^2-315*n^3+63*n^5)/576
%e A191685 c=7; a(n+1) = binomial(n+1,8)*(80*n+114*n^2-23*n^3-75*n^4-9*n^5+9*n^6)/144
%e A191685 c=8; a(n+1) = binomial(n+1,9)*(-1152-1936*n+2820*n^2+
%e A191685   5320*n^3+735*n^4-1575*n^5-315*n^6+135*n^7)/3840
%e A191685 c=9; a(n+1) = binomial(n+1,10)*(-1008*n-1676*n^2 +100*n^3+1295*n^4+392*n^5-210*n^6-60*n^7 +15*n^8)/768
%p A191685 I: programs generate the sequence:
%p A191685 with(combinat): c:=7; a:= proc(n) a(n):=abs(stirling1(n,n-c)); end: seq(a(n), n=c+1..28);
%p A191685 for n from 7 to 27 do a(n+1) := binomial(n+1,8)*(80*n+ 114*n^2- 23*n^3- 75*n^4- 9*n^5+ 9*n^6)/144 end do: seq(a(n),n=8..28);
%p A191685 II: program generates explicit formulas for a(n+1) =  s(n+1,n+1-c):
%p A191685 k[1,0]:=1: v:=1:
%p A191685 for c from 2 to 10 do
%p A191685   c1:=c-1: c2:=c-2: p0:=0:
%p A191685   for j from 0 to c2 do p0:=p0+k[c1,j]*m^j: end do:
%p A191685   f:=expand(2*c*(m+1)*p0/v):
%p A191685   p1:=0: p2:=0:
%p A191685   for j from 0 to c1 do
%p A191685     p1:=p1+k[c,j]*(m+1)^j:
%p A191685     p2:=p2+k[c,j]*m^j:
%p A191685   end do:
%p A191685   g:=collect((m+2)*p1-(m-c1)*p2-f,m):
%p A191685   kh[0]:=rem(g,m,m): Mk:=[kh[0]]: Mv:=[k[c,0]]:
%p A191685   for j from 1 to c1 do
%p A191685     kh[j]:=coeff(g,m^j):
%p A191685     Mk:=[op(Mk),kh[j]]: Mv:=[k[c,j],op(Mv)]:
%p A191685   end do:
%p A191685   sol:=solve(Mk,Mv):
%p A191685   v:=1:
%p A191685   for j from 1 to c do
%p A191685     k[c,c-j]:=eval(k[c,c-j],sol[1,j]):
%p A191685     nen[j]:=denom(k[c,c-j]):
%p A191685     v:=ilcm(v,nen[j]):
%p A191685   end do:
%p A191685   for j from 0 to c1 do k[c,j]:=k[c,j]*v:
%p A191685     printf("%8d",k[c,j]): end do:
%p A191685   p3:=0:
%p A191685   for j from 0 to c1 do p3:=p3+k[c,j]*n^j: end do:
%p A191685   s[n+1,n+1-c]:=binomial(n+1,c+1)*(c+1)*p3/(2^c*k[c,c1]):
%p A191685 end do:
%p A191685 for c from 2 to 10 do print("%a\n",s[n+1,n+1-c]):
%p A191685 end do:
%Y A191685 Cf. A130534, A000012 (c=0; 1st diagonal), A000217 (c=1; 2nd diagonal), A000914 (c=2; 3rd diagonal), A001303 (c=3; 4th diagonal), A000915 (c=4; 5th diagonal), A053567 (c=5; 6th diagonal), A112002 (c=6; 7th diagonal), A191685 (c=7; 8th diagonal).
%K A191685 nonn,easy
%O A191685 8,1
%A A191685 _Paul Weisenhorn_, Jun 11 2011