A210472 -id:A210472 - cp's OEIS frontend

A262809 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 13, 1, 1, 1, 75, 409, 63, 1, 1, 1, 541, 23917, 16081, 321, 1, 1, 1, 4683, 2244361, 10681263, 699121, 1683, 1, 1, 1, 47293, 308682013, 14638956721, 5552351121, 32193253, 8989, 1, 1, 1, 545835, 58514835289, 35941784497263, 117029959485121, 3147728203035, 1538743249, 48639, 1, 1

Offset: 0

Alois P. Heinz, Oct 02 2015

Also, A(n,k) is the number of alignments for k sequences of length n each (Slowinski 1998).
Row r > 0 is asymptotic to sqrt(r*Pi) * (r^(r-1)/(r-1)!)^n * n^(r*n+1/2) / (2^(r/2) * exp(r*n) * (log(2))^(r*n+1)), or equivalently to sqrt(r) * (r^(r-1)/(r-1)!)^n * (n!)^r / (2^r * (Pi*n)^((r-1)/2) * (log(2))^(r*n+1)). - Vaclav Kotesovec, Mar 23 2016
From Vaclav Kotesovec, Mar 23 2016: (Start)
Column k > 0 is asymptotic to sqrt(c(k)) * d(k)^n / (Pi*n)^((k-1)/2), where c(k) and d(k) are roots of polynomial equations of degree k, independent on n.
---------------------------------------------------
k               d(k)
---------------------------------------------------
2              5.8284271247461900976033774484193...
3             56.9476283720414911685286267804411...
4            780.2794068067951456595241495989622...
5          13755.2719024115081712083954421541320...
6         296476.9162644200814909862281498491264...
7        7553550.6198338218721069097516499501996...
8      222082591.6017202421029000117685530884167...
9     7400694480.0494436216324852038000444393262...
10  275651917450.6709238286995776605620357737005...
---------------------------------------------------
d(k) is a root of polynomial:
---------------------------------------------------
k=2, 1 - 6*d + d^2
k=3, -1 + 3*d - 57*d^2 + d^3
k=4, 1 - 12*d - 218*d^2 - 780*d^3 + d^4
k=5, -1 + 5*d - 1260*d^2 - 3740*d^3 - 13755*d^4 + d^5
k=6, 1 - 18*d - 5397*d^2 - 123696*d^3 + 321303*d^4 - 296478*d^5 + d^6
k=7, -1 + 7*d - 24031*d^2 - 374521*d^3 - 24850385*d^4 + 17978709*d^5 - 7553553*d^6 + d^7
k=8, 1 - 24*d - 102692*d^2 - 9298344*d^3 + 536208070*d^4 - 7106080680*d^5 - 1688209700*d^6 - 222082584*d^7 + d^8
(End)
d(k) = (2^(1/k) - 1)^(-k). - David Bevan, Apr 07 2022
d(k) is asymptotic to (k/log(2))^k/sqrt(2). - David Bevan, Apr 07 2022
A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column. - Andrew Howroyd, Jan 23 2020

			A(2,2) = 13: [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,1),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)].
Square array A(n,k) begins:
  1, 1,    1,        1,             1,                   1, ...
  1, 1,    3,       13,            75,                 541, ...
  1, 1,   13,      409,         23917,             2244361, ...
  1, 1,   63,    16081,      10681263,         14638956721, ...
  1, 1,  321,   699121,    5552351121,     117029959485121, ...
  1, 1, 1683, 32193253, 3147728203035, 1050740615666453461, ...

Alois P. Heinz, Antidiagonals n = 0..48, flattened
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Columns: A000012 (k=0 and k=1), A001850 (k=2), A126086 (k=3), A263064 (k=4), A263065 (k=5), A263066 (k=6), A263067 (k=7), A263068 (k=8), A263069 (k=9), A263070 (k=10).
Rows: A000012 (n=0), A000670 (n=1), A055203 (n=2), A062208 (n=3), A062205 (n=4), A263061 (n=5), A263062 (n=6), A062204 (n=7), A263063 (n=8), A263071 (n=9), A263072 (n=10).
Main diagonal: A262810.
Cf. A210472, A225094, A227578, A227655, A229142, A229345, A263159, A316674.
Cf. A188392, A330942, A331461, A331637.

A:= (n, k)-> add(add((-1)^i*binomial(j, i)*
     binomial(j-i, n)^k, i=0..j), j=0..k*n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[, 0] =  1; A[n, k_] := Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jul 22 2016, after Alois P. Heinz *)

T(n,k) = {my(m=n*k); sum(j=0, m, binomial(j,n)^k*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))} \\ Andrew Howroyd, Jan 23 2020

A(n,k) = Sum_{j=0..k*n} Sum_{i=0..j} (-1)^i*C(j,i)*C(j-i,n)^k.
A(n,k) = Sum_{i >= 0} binomial(i,n)^k/2^(i+1). - Peter Bala, Jan 30 2018
A(n,k) = Sum_{j=0..n*k} binomial(j,n)^k * Sum_{i=j..n*k} (-1)^(i-j) * binomial(i,j). - Andrew Howroyd, Jan 23 2020

A263159 Number A(n,k) of lattice paths starting at {n}^k and ending when k or any component equals 0, using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 13, 1, 1, 1, 15, 157, 63, 1, 1, 1, 31, 2101, 5419, 321, 1, 1, 1, 63, 32461, 717795, 220561, 1683, 1, 1, 1, 127, 580693, 142090291, 328504401, 9763807, 8989, 1, 1, 1, 255, 11917837, 39991899123, 944362553521, 172924236255, 454635973, 48639, 1, 1

Offset: 0

Alois P. Heinz, Oct 11 2015

			Square array A(n,k) begins:
  1, 1,    1,       1,            1,                1, ...
  1, 1,    3,       7,           15,               31, ...
  1, 1,   13,     157,         2101,            32461, ...
  1, 1,   63,    5419,       717795,        142090291, ...
  1, 1,  321,  220561,    328504401,     944362553521, ...
  1, 1, 1683, 9763807, 172924236255, 7622403922836151, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0+1, 2-10 give: A000012, A001850, A115866, A263162, A263163, A263164, A263165, A263166, A263167, A263168.
Rows n=0-1 give: A000012, A255047.
Main diagonal gives A263160.
Cf. A210472, A225094, A227578, A227655, A229142, A229345, A262809.

s:= proc(n) option remember; `if`(n=0, {[]},
      map(x-> [[x[], 0], [x[], 1]][], s(n-1)))
    end:
b:= proc(l) option remember; `if`(l=[] or l[1]=0, 1,
       add((p-> `if`(p[1]<0, 0, `if`(p[1]=0, 1, b(p)))
       )(sort(l-x)), x=s(nops(l)) minus {[0$nops(l)]}))
    end:
A:= (n, k)-> b([n$k]):
seq(seq(A(n,d-n), n=0..d), d=0..10);

g[k_] := Table[Reverse[IntegerDigits[n, 2]][[;;k]], {n, 2^k+1, 2^(k+1)-1}];
b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[k]}]];
a[n_, k_] := If[n == 0 || k == 0 || k == 1, 1, b[Table[n, {k}]]];
Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 25 2020, after Alois P. Heinz in A115866 *)

A089759 Table T(n,k), 0<=k, 0<=n, read by antidiagonals, defined by T(n,k) = (k*n)! / (n!)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 24, 90, 20, 1, 1, 1, 120, 2520, 1680, 70, 1, 1, 1, 720, 113400, 369600, 34650, 252, 1, 1, 1, 5040, 7484400, 168168000, 63063000, 756756, 924, 1, 1, 1, 40320, 681080400, 137225088000, 305540235000, 11732745024, 17153136, 3432, 1, 1

Offset: 0

Philippe Deléham, Jan 08 2004; revised Jun 08 2005

T(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1. - Alois P. Heinz, May 06 2013

			Row n=0: 1, 1,   1,      1,           1,               1, ... A000012
Row n=1: 1, 1,   2,      6,          24,             120, ... A000142
Row n=2: 1, 1,   6,     90,        2520,          113400, ... A000680
Row n=3: 1, 1,  20,   1680,      369600,       168168000, ... A014606
Row n=4: 1, 1,  70,  34650,    63063000,    305540235000, ... A014608
Row n=5: 1, 1, 252, 756756, 11732745024, 623360743125120, ... A014609

Alois P. Heinz, Antidiagonals n = 0..32, flattened
T. Chappell, A. Lascoux, S. Ole Warnaar, W. Zudilin, Logarithmic and complex constant term identities, arXiv:1112.3130 [math.CO], 2012.

Cf. A000680, A014606, A014608, A014609, A000984, A187783 (transposed version).
Main diagonal gives A034841.
Cf. A210472, A225094.

T:= (n, k)-> (k*n)!/(n!)^k:
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Aug 16 2012

T[n_, k_] := (k*n)!/(n!)^k; Table[T[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 19 2015 *)

Corrected by Alois P. Heinz, Aug 16 2012

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.

Original entry on oeis.org

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

A225094 Number A(n,k) of lattice paths without interior points from {n}^k to {0}^k using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 6, 2, 0, 1, 1, 24, 54, 2, 0, 1, 1, 120, 1944, 384, 2, 0, 1, 1, 720, 99000, 132000, 2550, 2, 0, 1, 1, 5040, 6966000, 79716000, 8059800, 16506, 2, 0, 1, 1, 40320, 655678800, 78928416000, 57010275000, 471369024, 105840, 2, 0, 1

Offset: 0

Alois P. Heinz, Apr 27 2013

An interior point p = (p_1, ..., p_k) has k>0 components with 0

			A(n,0) = 1: [()].
A(0,k) = 1: [{0}^k].
A(1,1) = 1: [(1), (0)].
A(2,1) = 0, there is no path from (2) to (0) without interior points.
A(1,2) = 2: [(1,1), (0,1), (0,0)], [(1,1), (1,0), (0,0)].
A(1,3) = 6: [(1,1,1), (0,1,1), (0,0,1), (0,0,0)], [(1,1,1), (0,1,1), (0,1,0), (0,0,0)], [(1,1,1), (1,0,1), (0,0,1), (0,0,0)], [(1,1,1), (1,0,1), (1,0,0), (0,0,0)], [(1,1,1), (1,1,0), (0,1,0), (0,0,0)], [(1,1,1), (1,1,0), (1,0,0), (0,0,0)].
Square array A(n,k) begins:
  1, 1, 1,     1,         1,              1, ...
  1, 1, 2,     6,        24,            120, ...
  1, 0, 2,    54,      1944,          99000, ...
  1, 0, 2,   384,    132000,       79716000, ...
  1, 0, 2,  2550,   8059800,    57010275000, ...
  1, 0, 2, 16506, 471369024, 38606650125120, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0, 2-4 give: A000012, A040000, A060774, A225220.
Rows n=0-4 give: A000012, A000142, A071798(k) (for k>0), A225096, A225221.
Main diagonal gives: A225111.
Cf. A089759 (unrestricted paths), A210472, A262809, A263159.

b:= proc(n, l) option remember; local m; m:= nops(l);
      `if`(m=0 or l[m]=0, 1, `if`(l[1]>0 and l[m] b(n, [n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, l_] := b[n, l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[l[[1]] > 0 && l[[m]] < n, 0, Sum[If[l[[i]] == 0, 0, b[n, Sort[ReplacePart[l, i -> l[[i]] - 1]]]], {i, 1, m}]]] ]; a[n_, k_] := b[n, Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A093964 a(n) = Sum_{k=1..n} kk!C(n,k).

Original entry on oeis.org

0, 1, 6, 33, 196, 1305, 9786, 82201, 767208, 7891281, 88776910, 1085051121, 14322674796, 203121569833, 3080677142466, 49764784609065, 853110593298256, 15469738758475041, 295858753755835158, 5951981987323272001, 125652953065713520020, 2777591594084193600441

Offset: 0

Ralf Stephan, Apr 20 2004

nonn

Limit to which the columns of array A093966 converge.
Number of objects in all permutations of n objects taken 1,2,...,n at a time. Example: a(2)=6 because the permutations of {a,b} taken 1 and 2 at a time are: a,b,ab and ba, containing altogether 1+1+2+2=6 objects. a(n)=Sum(k*A008279(n,k),k=1..n). - Emeric Deutsch, Aug 16 2006
The number of sequences -where each member is an element in a set consisting of n elements- such that the last member is a repetition of a former member. Example: Set of possible members: {l,r}. Sequences such that the last member is a repetition of a former member: l,l; r,r; l,r,l; l,r,r; r,l,l; r,l,r. a(n)=Sum(k*A008279(n,k),k=1..n). [From Franz Fritsche (ff(AT)simple-line.de), Feb 22 2009]
The total number of elements in all ascending runs (including runs of length 1) over all permutations of {1,2,...,n}.  a(2) = 6 because in the permutations [1,2] and [2,1] there are 4 runs of length 1 and 1 run of length 2. a(n) = Sum_{k>=1} A132159(n,k)*k. - Geoffrey Critzer, Feb 24 2014

			G.f. = x + 6*x^2 + 33*x^3 + 196*x^4 + 1305*x^5 + 9786*x^6 + 82201*x^7 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000522, A001339, A007526, A008279.

Row n=2 of A210472. - Alois P. Heinz, Jan 23 2013

[0] cat [n le 2 select 6^(n-1) else n*((n+1)*Self(n-1) - (n-1)*Self(n-2))/(n-1): n in [1..30]]; // G. C. Greubel, Dec 29 2021

seq(add(k*n!/(n-k)!,k=1..n),n=0..20); # Emeric Deutsch, Aug 16 2006
# second Maple program:
a:= proc(n) a(n):=`if`(n<2, n, n*((n+1)/(n-1)*a(n-1)-a(n-2))) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 21 2013

nn=21;Range[0,nn]!CoefficientList[Series[D[Exp[y x]/(1-x)^2,y]/.y->1,{x,0,nn}],x] (* Geoffrey Critzer, Feb 24 2014 *)

PARI
```
a(n)=sum(k=1,n,k*k!*binomial(n,k))
```

[factorial(n)*( x*exp(x)/(1-x)^2 ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Dec 29 2021

E.g.f.: x*exp(x)/(1-x)^2. - Vladeta Jovovic, Apr 24 2004
a(n) = 1 + (n-1)*floor(e*n!) = 1 + (n-1)*A000522(n) = A000522(n+1) - 2*A000522(n) = A001339(n) - A000522(n). - Henry Bottomley, Dec 22 2008
a(n) = n if n < 2, a(n) = n*((n+1)/(n-1)*a(n-1) - a(n-2)) for n >= 2. - Alois P. Heinz, Jan 21 2013
E.g.f.: x*(1- 12*x/(Q(0)+6*x-3*x^2))/(1-x)^2, where Q(k) = 2*(4*k+1)*(32*k^2+16*k+x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
G.f.: conjecture: T(0)/x - 1/x, where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
a(n) = n*a(n-1) + A007526(n), a(0) = 0. - David M. Cerna, May 12 2014

a(0) inserted by Alois P. Heinz, Jan 21 2013

A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 7, 22, 4, 1, 1, 25, 248, 188, 8, 1, 1, 121, 6506, 11380, 1712, 16, 1, 1, 721, 292442, 2359348, 577124, 16098, 32, 1, 1, 5041, 19450082, 1088626684, 991365512, 30970588, 154352, 64, 1

Offset: 0

Alois P. Heinz, Sep 24 2013

			A(2,2) = 22: [(2,2),(1,1),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)], [(2,2),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(1,2),(1,0),(0,0)], [(2,2),(0,2),(0,1),(0,0)], [(2,2),(0,2),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(0,1),(0,0)], [(2,2),(2,1),(2,0),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,0)], [(2,2),(2,0),(1,0),(0,0)], [(2,2),(2,0),(0,0)].
Square array A(n,k) begins:
  1,  1,     1,        1,            1,                 1, ...
  1,  1,     3,        7,           25,               121, ...
  1,  2,    22,      248,         6506,            292442, ...
  1,  4,   188,    11380,      2359348,        1088626684, ...
  1,  8,  1712,   577124,    991365512,     4943064622568, ...
  1, 16, 16098, 30970588, 453530591824, 25162900228200976, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0-3 give: A000012, A011782, A132595(n+1), A229482.
Rows n=0-2 give: A000012, A038507 (for k>1), A229465.
Main diagonal gives: A229346.
Cf. A060854, A227578, A227655, A225094, A210472, A229142, A262809, A263159.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`(m=0 or l[m]=0, 1,
      `if`(m>1, add(b(l-[j$m]), j=1..l[1]), 0)+
      add(add(b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..m))
    end:
A:= (n, k)-> b([n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Sep 24 2013

b[l_] := b[l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[m > 1, Sum[b[l - Array[j&, m]], {j, 1, l[[1]]}],  0] + Sum[Sum[b[Sort[ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, m}]]]; a[n_, k_] := b[Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A209245 Main diagonal of the triple recurrence x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) with x(i,j,k) = 1 if 0 in {i,j,k}.

Original entry on oeis.org

1, 3, 33, 543, 10497, 220503, 4870401, 111243135, 2602452993, 61985744967, 1497148260033, 36566829737727, 901314269530113, 22385640256615743, 559574590912019457, 14065064484334380543, 355222860485671141377, 9008982166319523972903, 229325469394627488082497

Offset: 0

Jon Perry, Jan 13 2013

nonn

Level sums are defined as the sum of x(i,j,k) with i,j,k >= 0 and i+j+k = n. This gives 3*A164039(n-1) for n>0.
Slice x(1,j,k) with j,k >= 0 of the cube begins:
  1,  1,  1,   1,   1,    1,    1,    1, ... A000012
  1,  3,  5,   7,   9,   11,   13,   15, ... A005408
  1,  5, 11,  19,  29,   41,   55,   71, ... A028387
  1,  7, 19,  39,  69,  111,  167,  239, ... A108766(k+1)
  1,  9, 29,  69, 139,  251,  419,  659, ...
  1, 11, 41, 111, 251,  503,  923, 1583, ...
  1, 13, 55, 167, 419,  923, 1847, 3431, ...
  1, 15, 71, 239, 659, 1583, 3431, 6863, ...
The main diagonal of the slice is A134760.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A164039, A134760, A209288.

Column k=3 of A210472. - Alois P. Heinz, Jan 23 2013

a:= proc(n) option remember; `if`(n<2, 2*n+1,
      ((888-3020*n+3668*n^2-1912*n^3+364*n^4) *a(n-1)
       +3*(3*n-4)*(7*n-5)*(2*n-3)*(3*n-5) *a(n-2)) /
       ((2*n-1)*(7*n-12)*(n-1)^2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 17 2013

b[] = 0; b[args__] := b[args] = If[{args}[[1]] == 0, 1, Sum[b @@ Sort[ ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]];
a[n_] := b @@ Table[n, 3];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 03 2018, from Alois P. Heinz's Maple code for A210472 *)

a(n) = x(n,n,n) with x(i,j,k) = 1 if 0 in {i,j,k} and x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) else.

a(n) ~ 3^(3*n+1/2) / (8*Pi*n). - Vaclav Kotesovec, Sep 07 2014

A209288 Main diagonal of the quadruple recurrence x(i,j,k,m) = x(i-1,j,k,m) + x(i,j-1,k,m) + x(i,j,k-1,m) + x(i,j,k,m-1) with x(i,j,k,m) = 1 if 0 in {i,j,k,m}.

Original entry on oeis.org

1, 4, 196, 22096, 3323092, 574346824, 107697153304, 21304602938056, 4376897152490644, 924871720044550888, 199731547307306769736, 43887077830441507774336, 9780481173520567895278600, 2205358814500087896152369104, 502225405515985555630557626848

Offset: 0

Jon Perry, Jan 16 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..417

Cf. A209245.

Column k=4 of A210472. - Alois P. Heinz, Jan 23 2013

b:= proc() option remember; `if`(args[1]=0, 1,
       add(b(sort(subsop(i=args[i]-1, [args]))[]), i=1..nargs))
    end:
a:= n-> b(n$4):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 18 2013

b[] = 0; b[args__] := b[args] = If[{args}[[1]] == 0, 1, Sum[b @@ Sort[ ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]];
a[n_] := b @@ Table[n, 4];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 03 2018, after Alois P. Heinz *)

a(n) = x(n,n,n,n) with x(i,j,k,m) = 1 if 0 in {i,j,k,m} and x(i,j,k,m) = x(i-1,j,k,m) + x(i,j-1,k,m) + x(i,j,k-1,m) + x(i,j,k,m-1) else.
a(n) ~ 2^(8*n-1/2) / (27*(Pi*n)^(3/2)). - Vaclav Kotesovec, Sep 07 2014
Recurrence: 2*(n-2)*(n-1)^3*(3*n - 4)*(3*n - 2)*(10773*n^5 - 127620*n^4 + 601635*n^3 - 1410376*n^2 + 1643420*n - 761136)*a(n) = (n-2)*(50320683*n^10 - 922567239*n^9 + 7517570148*n^8 - 35838081882*n^7 + 110640905811*n^6 - 231017836827*n^5 + 330199460678*n^4 - 318795408964*n^3 + 198794448664*n^2 - 72220580288*n + 11590694016)*a(n-1) - 2*(2*n - 3)*(43339779*n^10 - 841711662*n^9 + 7268645808*n^8 - 36726190830*n^7 + 120139923393*n^6 - 265623988980*n^5 + 401575152460*n^4 - 409434087632*n^3 + 269059885664*n^2 - 102737317696*n + 17273392896)*a(n-2) - 16*(2*n - 5)*(2*n - 3)*(3*n - 8)*(3*n - 7)*(4*n - 11)*(4*n - 9)*(10773*n^5 - 73755*n^4 + 198885*n^3 - 263461*n^2 + 170958*n - 43304)*a(n-3). - Vaclav Kotesovec, Sep 12 2016

A210486 Number of paths starting at {3}^n to a border position where one component equals 0 using steps that decrement one component by 1.

Original entry on oeis.org

0, 1, 20, 543, 22096, 1304045, 106478916, 11545342795, 1608000044288, 280061940550041, 59677171216017940, 15278632095285640631, 4628964787172536267920, 1638318264614752659427333, 669895681115518466689138436, 313418973409285344224352078435

Offset: 0

Alois P. Heinz, Jan 23 2013

			a(1) = 1: [3, 2, 1, 0].
a(2) = 20: [33, 23, 13, 03], [33, 23, 13, 12, 02], [33, 23, 13, 12, 11, 01], [33, 23, 13, 12, 11, 10], [33, 23, 22, 12, 02], [33, 23, 22, 12, 11, 01], [33, 23, 22, 12, 11, 10], [33, 23, 22, 21, 11, 01], [33, 23, 22, 21, 11, 10], [33, 23, 22, 21, 20], [33, 32, 22, 12, 02], [33, 32, 22, 12, 11, 01], [33, 32, 22, 12, 11, 10], [33, 32, 22, 21, 11, 01], [33, 32, 22, 21, 11, 10], [33, 32, 22, 21, 20], [33, 32, 31, 21, 11, 01], [33, 32, 31, 21, 11, 10], [33, 32, 31, 21, 20], [33, 32, 31, 30].

Alois P. Heinz, Table of n, a(n) for n = 0..237

Row n=3 of A210472.

a:= proc(n) option remember; `if`(n<3, [0, 1, 20][n+1],
      ((n-1)*(n-2)*(n+1)*a(n-3) -(n-1)*(3*n^2-2*n-4)*a(n-2)
      +(2*n+1)*(n^2-n+2)*a(n-1)) / (n-1))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 23 2013

a[n_] := a[n] = If[n<3, {0, 1, 20}[[n+1]], ((n-1)*(n-2)*(n+1)*a[n-3] - (n-1)*(3*n^2 - 2*n - 4)*a[n-2] + (2*n+1)*(n^2 - n + 2)*a[n-1]) / (n-1)];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 29 2017, after Alois P. Heinz *)

a(n) ~ sqrt(Pi) * 2^(n+1) * n^(2*n+3/2) / exp(2*n-1). - Vaclav Kotesovec, Sep 02 2014

Showing 1-10 of 11 results. Next

cp's OEIS Frontend

A262809 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A263159 Number A(n,k) of lattice paths starting at {n}^k and ending when k or any component equals 0, using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A089759 Table T(n,k), 0<=k, 0<=n, read by antidiagonals, defined by T(n,k) = (k*n)! / (n!)^k.

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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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A225094 Number A(n,k) of lattice paths without interior points from {n}^k to {0}^k using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A093964 a(n) = Sum_{k=1..n} k*k!*C(n,k).

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A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A093964 a(n) = Sum_{k=1..n} kk!C(n,k).