A229142 -id:A229142 - cp's OEIS frontend

A229676 a(n) = Sum_{k = 0..n} Product_{j = 0..8} C(n+j*k,k).

1, 362881, 12504639772801, 1080492192338314694401, 140810184334251776225321193601, 23183593018924832394604719137184142081, 4439414110286267003192333763481728593177802241, 944848564471993704169724618186222285154304912036663681

Offset: 0

Alois P. Heinz, Sep 27 2013

Number of lattice paths from {n}^9 to {0}^9 using steps that decrement one component or all components by 1.

Alois P. Heinz, Table of n, a(n) for n = 0..100

Column k = 9 of A229142.

with(combinat):
a:= n-> add(multinomial(n+8*k, n-k, k$9), k=0..n):
seq(a(n), n=0..10);

multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := Sum[multinomial[n + 8*k, Join[{n - k}, Array[k&, 9]]], {k, 0, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

a(n) = Sum_{k = 0..n} multinomial(n+8*k; n-k, {k}^9).
G.f.: Sum_{k >= 0} (9*k)!/k!^9 * x^k / (1-x)^(9*k+1).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 181441*x^2 + 4168213439041*x^3 + 270123052269252349441*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016

A229677 a(n) = Sum_{k = 0..n} Product_{j = 0..9} C(n+j*k,k).

Original entry on oeis.org

1, 3628801, 2375880907276801, 4386797386179342934060801, 12868640117405297821759744777996801, 49120459033702373637913562847507823210617601, 222254155614179529476178258638452174287098861960755201, 1132660294172702489573582429384603543633942385302181948349459201

Offset: 0

Alois P. Heinz, Sep 27 2013

Number of lattice paths from {n}^10 to {0}^10 using steps that decrement one component or all components by 1.

Alois P. Heinz, Table of n, a(n) for n = 0..100

Column k = 10 of A229142.

with(combinat):
a:= n-> add(multinomial(n+9*k, n-k, k$10), k=0..n):
seq(a(n), n=0..10);

multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := Sum[multinomial[n + 9*k, Join[{n - k}, Array[k&, 10]]], {k, 0, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

a(n) = Sum_{k = 0..n} multinomial(n+9*k; n-k, {k}^10).
G.f.: Sum_{k >= 0} (10*k)!/k!^10 * x^k / (1-x)^(10*k+1).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 1814401*x^2 + 791960304240001*x^3 + 1096699347338442061435201*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016

A336169 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * multinomial(n+(k-1)*j; n-j, {j}^k).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 5, 1, 0, 1, 1, 23, 67, 1, 0, 0, 1, 119, 2401, 1109, 1, 0, 1, 1, 719, 112681, 347279, 20251, 1, 0, 0, 1, 5039, 7479361, 166923119, 58370761, 391355, 1, 0, 1, 1, 40319, 681040081, 137127810959, 302857024681, 10693893503, 7847155, 1, 0, 0, 1, 362879, 81729285121, 182499151015439, 3244063941457921, 616967236620839, 2071837562929, 161476565, 1, 0, 1

Offset: 0

Seiichi Manyama, Jul 10 2020

Column k is the diagonal of the rational function 1 / (1 - Sum_{j=1..k} x_j + Product_{j=1..k} x_j) for k>0.

			Square array begins:
  1, 1, 1,      1,           1,               1, ...
  0, 0, 1,      5,          23,             119, ...
  1, 0, 1,     67,        2401,          112681, ...
  0, 0, 1,   1109,      347279,       166923119, ...
  1, 0, 1,  20251,    58370761,    302857024681, ...
  0, 0, 1, 391355, 10693893503, 616967236620839, ...

Columns k=0-5 give: A059841, A000007, A000012, A124435, A336170, A336171.
Rows n=0-1 give: A000012, A033312.
Main diagonal gives A336172.
Cf. A229142.

T[n_, k_] := Sum[(-1)^(n - j)*(n + (k - 1)*j)!/(n - j)!/(j!)^k, {j, 0, n} ]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 10 2020 *)

G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1+x)^(k*j+1).

cp's OEIS Frontend

A229676 a(n) = Sum_{k = 0..n} Product_{j = 0..8} C(n+j*k,k).

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A229677 a(n) = Sum_{k = 0..n} Product_{j = 0..9} C(n+j*k,k).

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A336169 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * multinomial(n+(k-1)*j; n-j, {j}^k).

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