A089465
3rd hyperbinomial transform of A001858; also the hyperbinomial transform of A089462.
Original entry on oeis.org
1, 4, 23, 178, 1763, 21504, 313585, 5342068, 104376201, 2304582544, 56807530871, 1547599725720, 46202052688603, 1500629138909632, 52697989385197137, 1990117967149595824, 80440669725095395025, 3465573101368534916928
Offset: 0
-
Table[Sum[Sum[Binomial[m, j]*Binomial[n, n - m - j + 1]*(n + 3)^(n - m - j + 1)*(m + j)!/(-2)^j, {j, 0, m}]/m!, {m, 0, n + 1}], {n, 0, 50}] (* G. C. Greubel, Nov 18 2017 *)
-
a(n)=if(n<0,0,sum(m=0,n+1,sum(j=0,m,binomial(m,j)*binomial(n,n-m-j+1)*(n+3)^(n-m-j+1)*(m+j)!/(-2)^j)/m!))
A089460
Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.
Original entry on oeis.org
1, 2, 1, 8, 4, 1, 50, 24, 6, 1, 432, 200, 48, 8, 1, 4802, 2160, 500, 80, 10, 1, 65536, 28812, 6480, 1000, 120, 12, 1, 1062882, 458752, 100842, 15120, 1750, 168, 14, 1, 20000000, 8503056, 1835008, 268912, 30240, 2800, 224, 16, 1, 428717762, 180000000, 38263752, 5505024, 605052, 54432, 4200, 288, 18, 1
Offset: 0
Rows begin:
{1},
{2,1},
{8,4,1},
{50,24,6,1},
{432,200,48,8,1},
{4802,2160,500,80,10,1},
{65536,28812,6480,1000,120,12,1},
{1062882,458752,100842,15120,1750,168,14,1},..
-
Join[{1}, Table[Binomial[n, k]*2*(n - k + 2)^(n - k - 1), {n, 1, 49}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 18 2017 *)
-
for(n=0,10, for(k=0,n, print1(2*(n-k+2)^(n-k-1)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 18 2017
A144304
Square array A(n,m), n>=0, m>=0, read by antidiagonals: A(n,m) = n-th number of the m-th iteration of the hyperbinomial transform on sequence A001858.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 3, 7, 7, 1, 4, 14, 38, 38, 1, 5, 23, 93, 291, 291, 1, 6, 34, 178, 822, 2932, 2932, 1, 7, 47, 299, 1763, 9193, 36961, 36961, 1, 8, 62, 462, 3270, 21504, 125292, 561948, 561948, 1, 9, 79, 673, 5523, 43135, 313585, 2022555, 10026505, 10026505, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, ...
2, 7, 14, 23, 34, ...
7, 38, 93, 178, 299, ...
38, 291, 822, 1763, 3270, ...
-
hymtr:= proc(p) proc(n,m) `if`(m=0, p(n), m*add(p(k) *binomial(n, k) *(n-k+m)^(n-k-1), k=0..n)) end end: f:= proc(n) option remember; add(add(binomial(m, j) *binomial(n-1, n-m-j) *n^(n-m-j) *(m+j)!/ (-2)^j/ m!, j=0..m), m=0..n) end: A:= hymtr(f): seq(seq(A(n, d-n), n=0..d), d=0..12);
-
hymtr[p_] := Function[{n, m}, If[m == 0, p[n], m*Sum[p[k]*Binomial[n, k]*(n-k+m)^(n-k-1), {k, 0, n}]]]; f[0] = 1; f[n_] := f[n] = Sum[Sum[Binomial[m, j]*Binomial[n-1, n-m-j]*n^(n-m-j)*(m+j)!/(-2)^j/m!, {j, 0, m}], {m, 0, n}]; A[0, ] = 1; A[1, k] := k+1; A[n_, m_] := hymtr[f][n, m]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
Showing 1-3 of 3 results.
Comments