A210391 -id:A210391 - cp's OEIS frontend

A005900 Octahedral numbers: a(n) = n(2n^2 + 1)/3.

0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, 19871, 21856, 23969, 26214, 28595, 31116, 33781, 36594, 39559, 42680

Offset: 0

N. J. A. Sloane

Series reversion of g.f.: A(x) is Sum_{n>0} - A066357(n)(-x)^n.
Partial sums of centered square numbers A001844. - Paul Barry, Jun 26 2003
Also as a(n) = (1/6)*(4n^3 + 2n), n>0: structured tetragonal diamond numbers (vertex structure 5) (cf. A000447 - structured diamonds); and structured trigonal anti-prism numbers (vertex structure 5) (cf. A100185 - structured anti-prisms). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Schlaefli symbol for this polyhedron: {3,4}.
If X is an n-set and Y and Z are disjoint 2-subsets of X then a(n-4) is equal to the number of 5-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
Starting with 1 = binomial transform of [1, 5, 8, 4, 0, 0, 0, ...] where (1, 5, 8, 4) = row 3 of the Chebyshev triangle A081277. - Gary W. Adamson, Jul 19 2008
a(n) = largest coefficient of (1 + ... + x^(n-1))^4. - R. H. Hardin, Jul 23 2009
Convolution square root of (1 + 6x + 19x^3 + ...) = (1 + 3x + 5x^2 + 7x^3 + ...) = A005408(x). - Gary W. Adamson, Jul 27 2009
Starting with offset 1 = the triangular series convolved with [1, 3, 4, 4, 4, ...]. - Gary W. Adamson, Jul 28 2009
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral, and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010
Let b be any product of four different primes. Then the divisor lattice of b^n is of width a(n+1). - Jean Drabbe, Oct 13 2010
Arises in Bezdek's proof on contact numbers for congruent sphere packings (see preprint). - Jonathan Vos Post, Feb 08 2011
Euler transform of length 2 sequence [6, -2]. - Michael Somos, Mar 27 2011
a(n+1) is the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 2n. - Clark Kimberling, Mar 19 2012
a(n) is the number of semistandard Young tableaux over all partitions of 3 with maximal element <= n. - Alois P. Heinz, Mar 22 2012
Self convolution of the odd numbers. - Reinhard Zumkeller, Apr 04 2012
a(n) is the number of (w,x,y,z) with all terms in {1,...,n} and w+x=y+z; also the number of (w,x,y,z) with all terms in {0,...,n} and |w-x|<=y. - Clark Kimberling, Jun 02 2012
The sequence is the third partial sum of (0, 1, 3, 4, 4, 4, ...). - Gary W. Adamson, Sep 11 2015
a(n) is the number of join-irreducible elements in the Weyl group of type B_n with respect to the strong Bruhat order. - Rafael Mrden, Aug 26 2020
Number of unit octahedra contained in an n-scale octahedron composed of a tetrahedral-octahedral honeycomb. The number of unit tetrahedra in it is 8*A000292(n-1) = 4*(n^3 - n)/3. Also, the number of unit tetrahedra and unit octahedra contained in an n-scale tetrahedron composed of a tetrahedral-octahedral honeycomb is respectively A006527(n) = (n^3 + 2*n)/3 and A000292(n-1) = (n^3 - n)/6. - Jianing Song, Feb 24 2025

			G.f. = x + 6*x^2 + 19*x^3 + 44*x^4 + 85*x^5 + 146*x^6 + 231*x^7 + ...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 50.
H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Karoly Bezdek, Contact numbers for congruent sphere packings, arXiv:1102.1198 [math.MG], 2011.
Matteo Cavaleri and Alfredo Donno, Some degree and distance-based invariants of wreath products of graphs, arXiv:1805.08989 [math.CO], 2018.
Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq (5), m=2.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Muhammad Fatih Killik and Bünyamin Şahi̇n, Further Results on Level Matrix, Int'l J. Math. Combin. (2024) Vol 4, 68-74. See p. 71.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Hankyung Ko, Volodymyr Mazorchuk and Rafael Mrđen, Join operation for the Bruhat order and Verma modules, arXiv:2109.01067 [math.RT], 2021. See Remark 5.10 p. 19.
A. Lascoux and M.-P. Schützenberger, Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), #R27.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
J. K. Merikoski, R. Kumar and R. A. Rajput, Upper bounds for the largest eigenvalue of a bipartite graph, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Octahedral Number.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Sums of 2 consecutive terms give A001845. Cf. A001844.
1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Cf. A022521.
Cf. A081277.
Row n=3 of A210391. - Alois P. Heinz, Mar 22 2012
Cf. A005408.
Cf. A053676, A053677, A053678.
Cf. A002061.
Cf. A000292 (tetrahedral numbers), A000578 (cubes), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
Similar sequence: A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099193(m=7), A099195 (m=8), A099196 (m=9), A099197 (m=10).

a005900 n = sum $ zipWith (*) odds $ reverse odds
            where odds = take n a005408_list
a005900_list = scanl (+) 0 a001844_list
-- Reinhard Zumkeller, Jun 16 2013, Apr 04 2012

[n*(2*n^2+1)/3: n in [0..50]]; // Wesley Ivan Hurt, Sep 11 2015

I:=[0,1,6,19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Sep 12 2015

al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(3,n), n=0..100)];
A005900:=(z+1)**2/(z-1)**4; # Simon Plouffe in his 1992 dissertation
with(combinat): seq(fibonacci(4,2*n)/12, n=0..40); # Zerinvary Lajos, Apr 21 2008

Table[(2n^3+n)/3, {n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,6,19},50] (* Harvey P. Dale, Oct 10 2013 *)
CoefficientList[Series[x (1 + x)^2/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 12 2015 *)

makelist(n*(2*n^2+1)/3, n, 0, 20); /* Martin Ettl, Jan 07 2013 */

PARI
```
{a(n) = n*(2*n^2+1)/3};
```

concat([0],Vec(x*(1 + x)^2/(1 - x)^4 + O(x^50))) \\ Indranil Ghosh, Mar 16 2017

def a(n): return n*(2*n*n + 1)//3
print([a(n) for n in range(41)]) # Michael S. Branicky, Sep 03 2021

a(n) = 1^2 + 2^2 + ... + (n-1)^2 + n^2 + (n-1)^2 + ... + 2^2 + 1^2. - Amarnath Murthy, May 28 2001
G.f.: x * (1 + x)^2 / (1 - x)^4. a(n) = -a(-n) = (2*n^3 + n) / 3.
a(n) = ( ((n+1)^5-n^5) - (n^5-(n-1)^5) )/30. - Xavier Acloque, Oct 17 2003
a(n) is the sum of the products pq, where p and q are both positive and odd and p + q = 2n, e.g., a(4) = 7*1 + 5*3 + 3*5 + 1*7 = 44. - Jon Perry, May 17 2005
a(n) = 4*binomial(n,3) + 4*binomial(n,2) + binomial(n,1). - Mitch Harris, Jul 06 2006
a(n) = binomial(n+2,3) + 2*binomial(n+1,3) + binomial(n,3), (this pair generalizes; see A014820, the 4-cross polytope numbers).
Sum_{n>=1} 1/a(n) = 3*gamma + 3*Psi((I*(1/2))*sqrt(2)) - (1/2)*(3*I)*Pi*coth((1/2)*Pi*sqrt(2)) - (1/2)*(3*I)*sqrt(2) = A175577, where I=sqrt(-1). - Stephen Crowley, Jul 14 2009
a(n) = A035597(n)/2. - J. M. Bergot, Jun 11 2012
a(n) = A000578(n) - 2*A000292(n-1) for n>0. - J. M. Bergot, Apr 05 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Wesley Ivan Hurt, Sep 11 2015
E.g.f.: (1/3)*x*(3 + 6*x + 2*x^2)*exp(x). - Ilya Gutkovskiy, Mar 16 2017
a(n) = (A002061(A002061(n+1)) - A002061(A002061(n)))/6. - Daniel Poveda Parrilla, Jun 10 2017
a(n) = 6*a(n-1)/(n-1) + a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018
Sum_{n >= 1} (-1)^(n+1)/(a(n)*a(n+1)) = 6*log(2) - 4 = 1/(6 + 2/(6 + 6/(6 + ... + n*(n-1)/(6 + ...)))). See A142983. - Peter Bala, Mar 06 2024

A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 8, 1, 6, 12, 16, 6, 32, 32, 28, 1, 64, 16, 128, 24, 96, 80, 256, 8, 44, 192, 22, 80, 512, 96, 1024, 1, 288, 448, 224, 30, 2048, 1024, 800, 40, 4096, 400, 8192, 240, 168, 2304, 16384, 10, 360, 204, 2112, 672, 32768, 68, 832, 160, 5376, 5120

Offset: 1

Gus Wiseman, Feb 14 2018

nonn

A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 6 tableaux:
1 3   1 2   1 2   1 2   1 1   1 1
2 4   3 4   3 3   2 3   2 3   2 2

Richard P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press, 1999, Chapter 7.10.

FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A191714, A210391, A228125, A296150, A296560, A296561, A299202, A299966, A300056, A300121.

conj[y_List]:=If[Length[y]===0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
conj[n_Integer]:=Times@@Prime/@conj[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]];
ssyt[n_]:=If[n===1,1,Sum[ssyt[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Rest[Divisors[n]]}]];
Table[ssyt[conj[n]],{n,50}]

Let b(n) = Sum_{d|n, d>1} b(n * d' / d) where if d = Product_i prime(s_i)^m(i) then d' = Product_i prime(s_i - 1)^m(i) and prime(0) = 1. Then a(n) = b(conj(n)) where conj = A122111.

A038163 G.f.: 1/((1-x)*(1-x^2))^3.

Original entry on oeis.org

1, 3, 9, 19, 39, 69, 119, 189, 294, 434, 630, 882, 1218, 1638, 2178, 2838, 3663, 4653, 5863, 7293, 9009, 11011, 13377, 16107, 19292, 22932, 27132, 31892, 37332, 43452, 50388, 58140, 66861, 76551, 87381, 99351, 112651, 127281

Offset: 0

N. J. A. Sloane

Number of symmetric nonnegative integer 6 X 6 matrices with sum of elements equal to 4*n, under action of dihedral group D_4. - Vladeta Jovovic, May 14 2000
Equals the triangular sequence convolved with the aerated triangular sequence, [1, 0, 3, 0, 6, 0, 10, ...]. - Gary W. Adamson, Jun 11 2009
Number of partitions of n (n>=1) into 1s and 2s if there are three kinds of 1s and three kinds of 2s. Example: a(2)=9 because we have 11, 11', 11", 1'1', 1'1", 1"1", 2, 2', and 2". - Emeric Deutsch, Jun 26 2009
Equals the tetrahedral numbers with repeats convolved with the natural numbers: (1 + x + 4x^2 + 4x^3 + ...) * (1 + 2x + 3x^2 + 4x^3 + ...) = (1 + 3x + 9x^2 + 19x^3 + ...). - Gary W. Adamson, Dec 22 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,0,3,-1).

Cf. A008619, A006918, A001753.
Cf. A096338.
Column k=3 of A210391. - Alois P. Heinz, Mar 22 2012
Cf. A000217.

import Data.List (inits, intersperse)
a038163 n = a038163_list !! n
a038163_list = map
    (sum . zipWith (*) (intersperse 0 $ tail a000217_list) . reverse) $
    tail $ inits $ tail a000217_list where
-- Reinhard Zumkeller, Feb 27 2015

G := 1/((1-x)^3*(1-x^2)^3): Gser := series(G, x = 0, 42): seq(coeff(Gser, x, n), n = 0 .. 37); # Emeric Deutsch, Jun 26 2009
# alternative
A038163 := proc(n)
    (4*n^5+90*n^4+760*n^3+2970*n^2+5266*n+3285+(-1)^n*(30*n^2+270*n+555))/3840 ;
end proc:
seq(A038163(n),n=0..30) ; # R. J. Mathar, Feb 22 2021

CoefficientList[Series[1/((1-x)*(1-x^2))^3, {x, 0, 40}], x] (* Jean-François Alcover, Mar 11 2014 *)
LinearRecurrence[{3,0,-8,6,6,-8,0,3,-1},{1,3,9,19,39,69,119,189,294},50] (* Harvey P. Dale, Nov 24 2022 *)

a(2*k) = (4*k + 5)*binomial(k + 4, 4)/5 = A034263(k); a(2*k + 1) = binomial(k + 4, 4)*(15 + 4*k)/5 = A059599(k), k >= 0.
a(n) = (1/3840)*(4*n^5 + 90*n^4 + 760*n^3 + 2970*n^2 + 5266*n + 3285 + (-1)^n*(30*n^2 + 270*n + 555)). Recurrence: a(n) = 3*a(n-1) - 8*a(n-3) + 6*a(n-4) + 6*a(n-5) - 8*a(n-6) + 3*a(n-8) - a(n-9). - Vladeta Jovovic, Apr 24 2002
a(n+1) - a(n) = A096338(n+2). - R. J. Mathar, Nov 04 2008

A209673 a(n) = count of monomials, of degree k=n, in the Schur symmetric polynomials s(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 1, 4, 19, 116, 751, 5552, 43219, 366088, 3245311, 30569012, 299662672, 3079276708, 32773002718, 362512238272, 4136737592323, 48773665308176, 591313968267151, 7375591544495636, 94340754464144215, 1237506718985945656, 16608519982801477908, 228013066931927465872

Offset: 0

Wouter Meeussen, Mar 11 2012

nonn

Main diagonal of triangle A191714.

a(n) is also the number of semistandard Young tableaux of size and maximal entry n. - Christian Stump, Oct 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Symmetric Polynomials
FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Cf. A191714, A209664, A209665, A209666, A209667, A209668, A209669, A209670, A209671, A209672, A209673.

Main diagonal of A210391. - Alois P. Heinz, Mar 22 2012

(* see A191714 *)
Tr /@ Table[(stanley[#, l] & /@ Partitions[l]), {l, 11}]
(* or *)
Table[SeriesCoefficient[1/((1-x)^(n*(n+1)/2) * (1+x)^(n*(n-1)/2)), {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Aug 06 2025 *)

a(12)-a(22) from Alois P. Heinz, Mar 11 2012

Typo in Mathematica program fixed by Vaclav Kotesovec, Mar 19 2015

A191714 a(n,k) equals the number of semistandard Young tableaux with shape of a partition of n and maximal element <= k.

Original entry on oeis.org

1, 1, 4, 1, 6, 19, 1, 9, 39, 116, 1, 12, 69, 260, 751, 1, 16, 119, 560, 1955, 5552, 1, 20, 189, 1100, 4615, 15372, 43219, 1, 25, 294, 2090, 10460, 40677, 131131, 366088, 1, 30, 434, 3740, 22220, 100562, 370909, 1168008, 3245311, 1, 36, 630, 6512, 45628, 239316, 1007083, 3570240, 11042199, 30569012, 1, 42, 882, 10868, 89420, 541926, 2596573, 10347864, 35587071, 108535130, 299662672, 1, 49, 1218, 17732, 170340, 1188341, 6466159, 28915056, 110426979, 370661885, 1117689232, 3079276708

Offset: 1

Wouter Meeussen, Jun 12 2011

Maximal element can be any integer, but is chosen here to be <=n.

			For n=3 and k=2 the SSYT are
par= {3}     SSYT= {{1, 1, 1}}, {{2, 1, 1}}, {{2, 2, 1}}, {{2, 2, 2}}
par= {2,1}   SSYT= {{2, 1}, {1}}, {{2, 2}, {1}}
par= {1,1,1} SSYT= none
counts 4+2+0 = 6 = a(3,2).
Table begins:
  1;
  1,  4;
  1,  6,  19;
  1,  9,  39,  116;
  1, 12,  69,  260,  751;
  1, 16, 119,  560, 1955,  5552;
  1, 20, 189, 1100, 4615, 15372, 43219; ...

Alois P. Heinz, Rows n = 1..44, flattened
R. Stanley, Hook Lengths and Contents

Cf. A102539, A210391.

Main diagonal gives A209673.

Needs["Combinatorica`"];
hooklength[(p_)?PartitionQ] := Block[{ferr = (PadLeft[1 + 0*Range[#1], Max[p]] &) /@ p}, DeleteCases[(Rest[FoldList[Plus, 0, #1]] &) /@ ferr + Reverse /@ Reverse[Transpose[(Rest[FoldList[Plus, 0, #1]] &) /@ Reverse[Reverse /@ Transpose[ferr]]]], 0, -1] - 1];
content[(p_)?PartitionQ]:= Block[{le= Max[p], ferr =(PadLeft[1+ 0*Range[#1], Max[p]]&) /@ p}, DeleteCases[ MapIndexed[-le+ Range[le,1,-1]- #1- Tr[#2]&, 0*ferr]*ferr,0,-1]+ le];
stanley[(p_)?PartitionQ, t_Integer] := Times @@ ((t + Flatten[content[p]])/Flatten[hooklength[p]]);
Table[Tr[ stanley[#,k]  &/@ Partitions[n] ] , {n,12}, {k,n}]

A138177 Triangle T(n,k) read by rows: number of k X k symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 7, 15, 10, 1, 10, 36, 52, 26, 1, 14, 74, 176, 190, 76, 1, 18, 132, 460, 810, 696, 232, 1, 23, 222, 1060, 2705, 3756, 2674, 764, 1, 28, 347, 2180, 7565, 15106, 17262, 10480, 2620, 1, 34, 525, 4204, 19013, 51162, 83440, 80816, 42732, 9496, 1, 40

Offset: 1

Vladeta Jovovic, Mar 03 2008

See the Brualdi/Ma reference for the connection to A161126. - Joerg Arndt, Nov 02 2014
T(n,k) is also the number of semistandard Young tableaux of size n whose entries span the interval 1..k. See also Gus Wiseman's comment in A138178. The T(4,2) = 7 semi-standard Young tableaux of size 4 spanning the interval 1..2 are:
   11  122  112  111  1222  1122  1112
   22  2    2    2                      . - Jacob Post, Jun 15 2018

			Triangle T(n,k) begins:
  1;
  1,  2;
  1,  4,   4;
  1,  7,  15,   10;
  1, 10,  36,   52,   26;
  1, 14,  74,  176,  190,   76;
  1, 18, 132,  460,  810,  696,  232;
  1, 23, 222, 1060, 2705, 3756, 2674, 764;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Richard A. Brualdi, Shi-Mei Ma, Enumeration of involutions by descents and symmetric matrices, European Journal of Combinatorics, vol.43, pp.220-228, (January 2015).
FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux
Samantha Dahlberg, Combinatorial Proofs of Identities Involving Symmetric Matrices, arXiv:1410.7356 [math.CO], (27-October-2014)

Cf. (row sums) A138178, A135589, A135588, A161126, A210391.
Main diagonal gives A000085. - Alois P. Heinz, Apr 06 2015
T(2n,n) gives A266305.
T(n^2,n) gives A268309.

gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Apr 06 2015

gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); A[n_, k_] := Coefficient[ Series [gf[k], {x, 0, n+1}], x, n]; T[n_, k_] := Sum[(-1)^j*Binomial[k, j]*A[n, k-j], {j, 0, k}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A210391(n,k-i). - Alois P. Heinz, Apr 06 2015

A181477 a(n) has generating function 1/((1-x)^k(1-x^2)^(k(k-1)/2)) for k=5.

Original entry on oeis.org

1, 5, 25, 85, 275, 751, 1955, 4615, 10460, 22220, 45628, 89420, 170340, 313140, 562020, 980628, 1676370, 2800410, 4596290, 7399930, 11732006, 18297950, 28155910, 42716750, 64037980, 94823756, 138922300, 201325900, 288988100

Offset: 0

Wouter Meeussen, Oct 24 2010

a(n-1,k) is conjectured to also be the count of monomials (or terms) in the Schur polynomials of k variables and degree n, summed over all partitions of n in at most k parts (zero-padded to length k).

			a(3)=85 since the Schur polynomial of 5 variables and degree 4 starts off as x[1]*x[2]*x[3]*x[4] + x[1]*x[2]*x[3]*x[5] + ... + x[4]*x[5]^3 + x[5]^4. The exponents collect to the padded partitions of 4 as 5*p(1) + 40*p(2) + 30*p(3) + 150*p(4) + 50*p(5) where p(1) is the lexicographically first padded partition of 4: {4,0,0,0}, a coded form of monomials x[i]^4, and p(5) stands for {1,1,1,1}, coding x[i]x[j]x[k]x[l] with all indices different.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Schur Polynomial

For k=2 (two variables): A002620, k=3: A038163, k=4: A054498 k=6: A181478, k=7: A181479, k=8: A181480.

Column k=5 of A210391. - Alois P. Heinz, Mar 22 2012

Tr[toz/@(Function[q,PadRight[q,k]]/@ (TransposePartition/@ Partitions[n,k]))/. w[arg__] -> 1 ]; with toz[p_]:=Block[{a,q,e,w}, u1=Expand[q Together[Expand[schur[p]]] +q a]/. Plus-> List ; u2=u1/. Times->w /. q->Sequence[]/. w[i_Integer, r__]-> i w[r] /. x[]^(e:1) ->e ; u3=Plus@@ u2/. w[arg__]:> Reverse@ Sort@ w[arg] /. w[a]->0 ]; and schur[p_]:=Block[{le=Length[p],n=Tr[p]}, Together[Expand[Factor[Det[Outer[ #2^#1&,p+le-Range[le] , Array[x,le]]]]/Factor[Det[Outer[ #2^#1&,Range[le-1,0,-1] , Array[x,le]]]] ]] ]

cp's OEIS Frontend

A005900 Octahedral numbers: a(n) = n*(2*n^2 + 1)/3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Magma

Maple

Mathematica

Maxima

PARI

PARI

Python

Formula

A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A038163 G.f.: 1/((1-x)*(1-x^2))^3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A209673 a(n) = count of monomials, of degree k=n, in the Schur symmetric polynomials s(mu,k) summed over all partitions mu of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A191714 a(n,k) equals the number of semistandard Young tableaux with shape of a partition of n and maximal element <= k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A138177 Triangle T(n,k) read by rows: number of k X k symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n, n>=1, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A181477 a(n) has generating function 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)) for k=5.

Views

Author

A005900 Octahedral numbers: a(n) = n(2n^2 + 1)/3.

A181477 a(n) has generating function 1/((1-x)^k(1-x^2)^(k(k-1)/2)) for k=5.

A181478 a(n) has generating function 1/((1-x)^k(1-x^2)^(k(k-1)/2)) for k=6.

A181479 a(n) has generating function 1/((1-x)^k(1-x^2)^(k(k-1)/2)) for k=7.