A092288 -id:A092288 - cp's OEIS frontend

A091298 Triangle read by rows: T(n,k) is the number of plane partitions of n containing exactly k parts.

1, 1, 2, 1, 2, 3, 1, 4, 3, 5, 1, 4, 7, 5, 7, 1, 6, 10, 13, 7, 11, 1, 6, 14, 20, 19, 11, 15, 1, 8, 18, 33, 32, 31, 15, 22, 1, 8, 25, 43, 56, 54, 43, 22, 30, 1, 10, 29, 66, 81, 99, 78, 64, 30, 42, 1, 10, 37, 83, 126, 150, 148, 118, 88, 42, 56, 1, 12, 44, 114, 174, 246, 235, 230, 166, 124, 56, 77

Offset: 1

Wouter Meeussen, Feb 24 2004

First column is 1, representing the single-part {{n}}, last column is P(n), since the all-ones plane partitions form the Ferrers-Young plots of the (linear) partitions of n.

A plane partition of n is a two-dimensional table (or matrix) with nonnegative elements summing up to n, and nonincreasing rows and columns. (Zero rows and columns are ignored.) - M. F. Hasler, Sep 22 2018

			This plane partition of n=7: {{3,1,1},{2}} contains 4 parts: 3,1,1,2.
Triangle T(n,k) begins:
  1;
  1,  2;
  1,  2,  3;
  1,  4,  3,  5;
  1,  4,  7,  5,  7;
  1,  6, 10, 13,  7, 11;
  1,  6, 14, 20, 19, 11, 15;
  1,  8, 18, 33, 32, 31, 15, 22;
  1,  8, 25, 43, 56, 54, 43, 22, 30;
  1, 10, 29, 66, 81, 99, 78, 64, 30, 42;
  ...

Alois P. Heinz, Rows n = 1..50
A. Rovenchak, Enumeration of plane partitions with a restricted number of parts, arXiv preprint arXiv:1401.4367 [math-ph], 2014.
E. W. Weisstein, Plane partition.
Wikipedia, Plane partition.

Row sums give A000219.
Cf. A090539, A092288, A089924.
Column 1 is A000012. Column 2 is A052928. Diagonal and subdiagonal are A000041.

(* see A089924 for "planepartition" *) Table[Length /@ Split[Sort[Length /@ Flatten /@ planepartitions[n]]], {n, 16}]

A091298(n,k)=sum(i=1,#n=PlanePartitions(n),sum(j=1,#n[i],#n[i][j])==k)
PlanePartitions(n,L=0,PP=List())={ n<2&&return([if(n,[[1]],[])]); for(N=1,n, my(P=apply(Vecrev, if(L, select(p->vecmin(L-Vecrev(p,#L))>=0, partitions(N,L[1],#L)), partitions(N)))); if(NM. F. Hasler, Sep 24 2018

A081105 5th binomial transform of (1,1,0,0,0,0,.....).

Original entry on oeis.org

1, 6, 35, 200, 1125, 6250, 34375, 187500, 1015625, 5468750, 29296875, 156250000, 830078125, 4394531250, 23193359375, 122070312500, 640869140625, 3356933593750, 17547607421875, 91552734375000, 476837158203125

Offset: 0

Paul Barry, Mar 07 2003

Main diagonal of array defined by m(1,j)=j; m(i,1)=i and m(i,j)=m(i-1,j)+4*m(i-1,j-1) - Benoit Cloitre, Jun 13 2003

Index entries for linear recurrences with constant coefficients, signature (10,-25).

Cf. A001792, A006234, A079027, A092288.

LinearRecurrence[{10,-25},{1,6},30] (* Harvey P. Dale, Jan 20 2014 *)

a(n)=if(n,([0,1; -25,10]^n*[1;6])[1,1],1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 10*a(n-1)-25*a(n-2), a(0)=1, a(1)=6.
a(n) = (n+5)*5^(n-1).
G.f.: (1-4x)/(1-5x)^2.
a(n) = A079027(n), n>0. - R. J. Mathar, Sep 18 2008
From Amiram Eldar, Jan 19 2021: (Start)
Sum_{n>=0} 1/a(n) = 15625*log(5/4) - 41825/12.
Sum_{n>=0} (-1)^n/a(n) = 15625*log(6/5) - 34175/12. (End)

A090539 Total number of parts equal to 1 in all plane partitions of n.

Original entry on oeis.org

0, 1, 4, 11, 28, 62, 137, 278, 561, 1080, 2051, 3778, 6885, 12255, 21589, 37409, 64160, 108612, 182234, 302461, 497997, 812579, 1316225, 2115608, 3378239, 5357855, 8447086, 13237631, 20632630, 31985504, 49339795, 75738099, 115731636, 176055280, 266697522

Offset: 0

Wouter Meeussen, Feb 01 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Column k=1 of A092288.

Cf. A066633, A000219, A091298.

Table[Count[Flatten[planepartitions[k]], 1], {k, 17}]

A090539(n)=A092288(n,1) \\ M. F. Hasler, Sep 26 2018

a(18)-a(27) from Vaclav Kotesovec, May 05 2018

a(28)-a(34) from Alois P. Heinz, Sep 24 2018

A319648 Total number of parts in all plane partitions of n.

Original entry on oeis.org

0, 1, 5, 14, 38, 85, 196, 401, 830, 1615, 3119, 5802, 10718, 19246, 34276, 59889, 103656, 176801, 299025, 499732, 828638, 1360696, 2218128, 3586194, 5759839, 9184715, 14557974, 22929745, 35916469, 55942850, 86695329, 133671740, 205144324, 313380895, 476667370

Offset: 0

Alois P. Heinz, Sep 25 2018

nonn

			The plane partitions of 2 are [2], [1 1] and [1; 1]. There is a total of a(2) = 5 parts. - _M. F. Hasler_, Sep 27 2018

Alois P. Heinz, Table of n, a(n) for n = 0..50

Row sums of A092288.

Cf. A000219.

A319648(n)={vecsum(apply(pp->vecsum(apply(p->#p,pp)),PlanePartitions(n)))} \\ See A091298 for PlanePartitions(). For illustration mainly, becomes slow for n > 15. - M. F. Hasler, Sep 27 2018

M319648=[]; A319648(n,L=0,s)={if(L, n>1||return([1,1]); #L>2||(s=setsearch(M319648,[[n,L],[]],1))>#M319648|| M319648[s][1]!=[n,L]|| return(M319648[s][2]); my(S=[1,n]); for(m=2,n, forpart(P=m, vecmin(L-Vecrev(P,#L))<0&&next; S+=if(mA319648(n-m,Vecrev(P))*[1,#P;0,1],[1,#P]),L[1],#L)); #L>2|| M319648=setunion(M319648,[[[n,L],S]]); S, my(S=n); n>1&& forpart(P=n,S+=#P); for(m=2,n-1,forpart(P=m,S+=A319648(n-m,Vecrev(P))*[#P,1]~));S)} \\ M. F. Hasler, Sep 30 2018

a(n) = Sum_{k=1..n} k*A091298(n,k). - M. F. Hasler, Sep 27 2018

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A091298 Triangle read by rows: T(n,k) is the number of plane partitions of n containing exactly k parts.

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A081105 5th binomial transform of (1,1,0,0,0,0,.....).

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A090539 Total number of parts equal to 1 in all plane partitions of n.

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A319648 Total number of parts in all plane partitions of n.

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