A229675 -id:A229675 - cp's OEIS frontend

A229142 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 13, 1, 1, 1, 25, 115, 63, 1, 1, 1, 121, 2641, 2371, 321, 1, 1, 1, 721, 114121, 392641, 54091, 1683, 1, 1, 1, 5041, 7489441, 169417921, 67982041, 1307377, 8989, 1, 1, 1, 40321, 681120721, 137322405361, 308238414121, 12838867105, 32803219, 48639, 1, 1

Offset: 0

Alois P. Heinz, Sep 23 2013

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>1. - Seiichi Manyama, Jul 10 2020

			A(1,3) = 3*2+1 = 7:
          (0,1,1)-(0,0,1)
         /       X       \
  (1,1,1)-(1,0,1) (0,1,0)-(0,0,0)
       \ \       X       / /
        \ (1,1,0)-(1,0,0) /
         `---------------´
Square array A(n,k) begins:
  1, 1,    1,       1,           1,               1, ...
  1, 1,    3,       7,          25,             121, ...
  1, 1,   13,     115,        2641,          114121, ...
  1, 1,   63,    2371,      392641,       169417921, ...
  1, 1,  321,   54091,    67982041,    308238414121, ...
  1, 1, 1683, 1307377, 12838867105, 629799991355641, ...

Alois P. Heinz, Antidiagonals n = 0..44, flattened

Columns k=0+1, 2-10 give: A000012, A001850, A081798, A082488, A082489, A229049, A229674, A229675, A229676, A229677.
Rows n=0-1 give: A000012, A038507 (for k>1).
Main diagonal gives: A229267.
Cf. A060854, A227578, A227655, A225094, A210472, A262809, A263159.

with(combinat):
A:= (n,k)-> `if`(k<2, 1, add(multinomial(n+(k-1)*j, n-j, j$k), j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

a[, 0] = a[, 1] = 1; a[n_, k_] := Sum[Product[Binomial[n+j*m, m], {j, 0, k-1}], {m, 0, n}]; Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

A(n,k) = Sum_{j=0..n} multinomial(n+(k-1)*j; n-j, {j}^k) for k>1, A(n,0) = A(n,1) = 1.

G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1-x)^(k*j+1). for k>1. - Seiichi Manyama, Jul 10 2020

A081798 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k).

Original entry on oeis.org

1, 7, 115, 2371, 54091, 1307377, 32803219, 844910395, 22188235867, 591446519797, 15953338537885, 434479441772845, 11927609772412075, 329653844941016785, 9163407745486783435, 255982736410338609931, 7181987671728091545787

Offset: 0

Emanuele Munarini, Apr 23 2003

a(n) is also a generalization of Delannoy numbers to 3D; i.e. the number of walks from (0,0,0) to (n,n,n) in a 3D square lattice where each step is in the direction of one of (1,0,0), (0,1,0), (0,0,1) and (1,1,1). - Theodore Kolokolnikov, Jul 04 2010

Diagonal of the rational function 1/(1 - x - y - z - x*y*z). - Gheorghe Coserea, Jul 06 2016

Alois P. Heinz, Table of n, a(n) for n = 0..500
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
E. W. Weisstein, in MathWorld: Multinomial Coefficient.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Column k = 3 of A229142. Cf. A001850, A082488, A082459, A229049, A229674, A229675, A229676, A229677.

Related to diagonal of rational functions: A268545-A268555.

w := proc(i,j,k) option remember; if i=0 and j=0 and k = 0 then 1; elif i<0 or j<0 or k<0 then 0 else w(i-1,j,k)+w(i,j-1,k)+w(i,j,k-1)+w(i-1,j-1,k-1); end: end: for k from 0 to 10 do lprint(w(k,k,k)):end: # Theodore Kolokolnikov, Jul 04 2010
# second Maple program:
a:= proc(n) option remember; `if`(n<3, 51*n^2-45*n+1,
     ((3*n-2)*(30*n^2-50*n+13)*a(n-1)+(3*n-1)*(n-2)^2*a(n-3)
     -(9*n^3-30*n^2+29*n-6)*a(n-2))/(n^2*(3*n-4)))
    end:
seq(a(n), n=0..20); # Alois P. Heinz, Sep 22 2013

f[n_] := Sum[ Binomial[n, k] Binomial[n + k, k] Binomial[n + 2k, k], {k, 0, n}]; Array[f, 17, 0] (* Robert G. Wilson v *)
CoefficientList[Series[HypergeometricPFQ[{1/3, 2/3}, {1}, 27*x/(1 - x)^3]/(1 - x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 07 2016 *)

makelist(sum(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k),k,0,n),n,0,12);

{a(n)=polcoeff(sum(m=0,n,(3*m)!/m!^3*x^m/(1-x+x*O(x^n))^(3*m+1)), n)} \\ Paul D. Hanna, Sep 22 2013

a(n) = sum(k = 0, n, binomial(n, k) * binomial(n+k, k) * binomial(n+2*k, k)); \\ Michel Marcus, Jan 14 2016

a(n) = w(n,n,n) where w(i,j,k)=w(i-1,j,k)+w(i,j-1,k)+w(i,j,k-1)+w(i-1,j-1,k-1) and where w(0,0,0)=1 and w(i,j,k)=0 if one of i,j,k is strictly negative. - Theodore Kolokolnikov, Jul 04 2010
G.f.: hypergeom([1/3, 2/3],[1],27*x/(1-x)^3)/(1-x). - Mark van Hoeij, Oct 24 2011
G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^n / (1-x)^(3*n+1). - Paul D. Hanna, Sep 22 2013
a(n) ~ c*d^n/(Pi*n), where d = (3*(292 + 4*sqrt(5))^(2/3) + 132 + 20*(292 + 4*sqrt(5))^(1/3)) / (2*(292 + 4*sqrt(5))^(1/3)) = 29.900786688498085... is the root of the equation -1 + 3*d - 30*d^2 + d^3 = 0 and c = 1/(2*sqrt(((81 - 27*sqrt(5))/2)^(1/3) + 3*((3 + sqrt(5))/2)^(1/3) - 6)) = 0.8959908650405192232... is the root of the equation -1 - 72*c^2 - 1296*c^4 + 1728*c^6 = 0. - Vaclav Kotesovec, Sep 23 2013, updated Jul 07 2016
From Peter Bala, Jan 13 2016: (Start)
a(n) = Sum_{k = 0..n} multinomial(n + 2*k, k, k, k, n - k). Cf. A001850(n) = Sum_{k = 0..n} multinomial(n + k, k, k, n - k).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 42*x^3 + 639*x^4 + 11571*x^5  + ... appears to have integer coefficients. (End)
Conjecture: n^2*(3*n-4)*a(n) -(3*n-2)*(30*n^2-50*n+13)*a(n-1) +(9*n^3-30*n^2+29*n-6)*a(n-2) -(3*n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Apr 15 2016
Conjecture: (n^2)*a(n) +(-28*n^2+24*n-3)*a(n-1) +3*(-19*n^2+78*n-77)*a(n-2) +(5*n-12)*(n-3)*a(n-3) -2*(n-3)^2*a(n-4)=0. - R. J. Mathar, Apr 15 2016
0 = (2*x+1)*(x^3-3*x^2+30*x-1)*x*y'' + (6*x^4-8*x^3+51*x^2+60*x-1)*y' + (x-1)*(2*x^2+2*x-7)*y, where y is g.f. - Gheorghe Coserea, Jul 06 2016

cp's OEIS Frontend

A229142 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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