OFFSET
8,1
COMMENTS
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.
REFERENCES
K. Seidel, Variation der Binomialkoeffizienten, Bild
der Wissenschaft, 6 (1980), pp. 127-128.
LINKS
T. D. Noe, Table of n, a(n) for n = 8..1000
FORMULA
a(n+1) = A130534(T(n,n-7)) = s(n+1,n+1-7)
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
EXAMPLE
c=1; a(n+1) = binomial(n+1,2)
c=2; a(n+1) = binomial(n+1,3)*(2+3*n)/4
c=3; a(n+1) = binomial(n+1,4)*(n+n^2)/2
c=4; a(n+1) = binomial(n+1,5)*(-8-10*n+15*n^2 +15*n^3)/48
c=5; a(n+1) = binomial(n+1,6)*(-6*n-7*n^2+2*n^3+ 3*n^4)/16
c=6; a(n+1) = binomial(n+1,7)*(96+140*n-224*n^2-315*n^3+63*n^5)/576
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
c=8; a(n+1) = binomial(n+1,9)*(-1152-1936*n+2820*n^2+
5320*n^3+735*n^4-1575*n^5-315*n^6+135*n^7)/3840
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
MAPLE
I: programs generate the sequence:
with(combinat): c:=7; a:= proc(n) a(n):=abs(stirling1(n, n-c)); end: seq(a(n), n=c+1..28);
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);
II: program generates explicit formulas for a(n+1) = s(n+1, n+1-c):
k[1, 0]:=1: v:=1:
for c from 2 to 10 do
c1:=c-1: c2:=c-2: p0:=0:
for j from 0 to c2 do p0:=p0+k[c1, j]*m^j: end do:
f:=expand(2*c*(m+1)*p0/v):
p1:=0: p2:=0:
for j from 0 to c1 do
p1:=p1+k[c, j]*(m+1)^j:
p2:=p2+k[c, j]*m^j:
end do:
g:=collect((m+2)*p1-(m-c1)*p2-f, m):
kh[0]:=rem(g, m, m): Mk:=[kh[0]]: Mv:=[k[c, 0]]:
for j from 1 to c1 do
kh[j]:=coeff(g, m^j):
Mk:=[op(Mk), kh[j]]: Mv:=[k[c, j], op(Mv)]:
end do:
sol:=solve(Mk, Mv):
v:=1:
for j from 1 to c do
k[c, c-j]:=eval(k[c, c-j], sol[1, j]):
nen[j]:=denom(k[c, c-j]):
v:=ilcm(v, nen[j]):
end do:
for j from 0 to c1 do k[c, j]:=k[c, j]*v:
printf("%8d", k[c, j]): end do:
p3:=0:
for j from 0 to c1 do p3:=p3+k[c, j]*n^j: end do:
s[n+1, n+1-c]:=binomial(n+1, c+1)*(c+1)*p3/(2^c*k[c, c1]):
end do:
for c from 2 to 10 do print("%a\n", s[n+1, n+1-c]):
end do:
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Jun 11 2011
STATUS
approved