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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 2
5040, 109584, 1172700, 8409500, 45995730, 206070150, 790943153, 2681453775, 8207628000, 23057159840, 60202693980, 147560703732, 342252511900, 756111184500, 1599718388730, 3256091103430, 6400590336096, 12191224980000, 22563937825000, 40681506808800 (list; graph; refs; listen; history; text; internal format)



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.


K. Seidel, Variation der Binomialkoeffizienten, Bild

der Wissenschaft, 6 (1980), pp. 127-128.


T. D. Noe, Table of n, a(n) for n = 8..1000


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


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+


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


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:


  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):


  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:


  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:


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).

Sequence in context: A061122 A029575 A179965 * A213877 A135456 A254080

Adjacent sequences:  A191682 A191683 A191684 * A191686 A191687 A191688




Paul Weisenhorn, Jun 11 2011



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Last modified December 6 01:46 EST 2021. Contains 349558 sequences. (Running on oeis4.)