A008763 -id:A008763 - cp's OEIS frontend

A082424 Coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n), where s(n,n) is the Schur function indexed by two parts of size n, s(2n) is the Schur function corresponding to the trivial representation and * represents the inner or Kronecker product.

1, 1, 11, 41, 320, 1917, 14582, 100562, 688427, 4380888, 26324611, 148136566, 785175771, 3925637781, 18586683128, 83578440418, 358079558873, 1465784048253, 5748270468573, 21649265291143, 78483868584001

Offset: 0

Mike Zabrocki, Apr 24 2003

nonn

I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Oxford Univ. Press, second edition, 1995.

Cf. A008763 change 6 to 4 in the above program.

compsclr := proc(k) local gamma; add( combinat[Chi]( [k,k], gamma)^6/ZEE(gamma),gamma= combinat[partition](2*k)); end: ZEE := proc (mu) local res, m, i; m := 1; res := convert(mu,`*`); for i from 2 to nops(mu) do if mu[i] <> mu[i-1] then m := 1 else m := m+1 fi; res := res*m; od; res; end:

a(n) = Sum_{gamma} Chi^{(n, n)}( gamma )^6/z(gamma) the sum is over all partitions gamma of 2n Chi^lambda(gamma) is the value of the symmetric group character z(gamma) is the size of the stablizer of the conjugacy class of symmetric group indexed by the partition gamma

A082437 Coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n), where s(2n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding two the two row partition and * represents the inner or Kronecker product of symmetric functions.

Original entry on oeis.org

1, 0, 5, 1, 36, 15, 228, 231, 1313, 1939, 6971, 11899, 33118, 59543, 140620, 254476, 538042, 959028, 1871808, 3258512, 5981444, 10140360, 17726166, 29257848, 49127549, 79032258, 128267727, 201437596

Offset: 0

Mike Zabrocki, Apr 25 2003

nonn

I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Oxford Univ. Press, second edition, 1995.

Cf. A008763 for Chi( [k, k], gamma)^4/ZEE(gamma) instead of Chi( [k, k], gamma)^5/ZEE(gamma) in the programs above.

compsclr := proc(k) local gamma; add( combinat[Chi]( [k,k], gamma)^5/ZEE(gamma),gamma= combinat[partition](2*k)); end: ZEE := proc (mu) local res, m, i; m := 1; res := convert(mu,`*`); for i from 2 to nops(mu) do if mu[i] <> mu[i-1] then m := 1 else m := m+1 fi; res := res*m; od; res; end:

a(n) = Sum_{gamma} Chi^{(n, n)}( gamma )^5/z(gamma) the sum is over all partitions gamma of 2n Chi^lambda(gamma) is the value of the symmetric group character z(gamma) is the size of the stablizer of the conjugacy class of symmetric group indexed by the partition gamma

A060962 Expansion of (1+x^2)*(1+x^5)/( Product_{j=1..7} (1-x^j) ).

Original entry on oeis.org

1, 1, 3, 4, 7, 11, 17, 25, 36, 50, 69, 93, 125, 163, 213, 272, 346, 435, 543, 671, 825, 1005, 1218, 1466, 1756, 2090, 2478, 2921, 3430, 4009, 4669, 5414, 6259, 7207, 8274, 9468, 10803, 12289, 13944, 15777, 17809, 20052, 22528, 25249, 28243, 31522, 35115, 39041, 43327, 47995, 53078

Offset: 0

N. J. A. Sloane, May 05 2002

G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis VIII: Plane partition diamonds, Advances Applied Math., 27 (2001), 231-242 (Cor. 2.1, n=2).

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A008763, A069950.

R:=PowerSeriesRing(Integers(), 55); Coefficients(R!( (1+x^2)*(1+x^5)/( &*[1-x^j: j in [1..7]] ) )); // G. C. Greubel, Jan 15 2020

seq(coeff(series((1+x^2)*(1+x^5)/(mul(1-x^j, j=1..7)), x, n+1), x, n), n = 0..55); # G. C. Greubel, Jan 15 2020

Table[SeriesCoefficient[(1+x^2)*(1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7),{x,0,n}],{n,0,55}] (* Vaclav Kotesovec, Oct 01 2012 *)

Vec((1+x^2)*(1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7) +O(x^55)) \\ Charles R Greathouse IV, Sep 27 2012

def A060962_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)*(1+x^5)/(product(1-x^j for j in (1..7))) ).list()
A060962_list(55) # G. C. Greubel, Jan 15 2020

a(n) = 1 - 263117*n/1814400 + 109537*n^2/907200 + 97*n^3/4320 + 127*n^4/72576 + n^5/14400 + n^6/907200 + 4/7*floor(n/7) + 1/3*floor(n/6) + 2/25*floor(n/5) - 11/162*floor(n/3) + (73/192 + 5*n/96)*floor(n/2) + 1/7*floor((1+n)/7) - 1/6*floor((1+n)/6) + 4/25*floor((1+n)/5) + (53/162 + n/54)*floor((1+n)/3) + 3/7*floor((2+n)/7) - 4/25*floor((2+n)/5) + 2/7*floor((3+n)/7) + 8/25*floor((3+n)/5) + 1/7*floor((4+n)/7) + 3/7*floor((5+n)/7). - Vaclav Kotesovec, Sep 29 2012

A089300 Number of planar partitions of n (A000219) that are non-squashing along rows and down columns (cf. A018819).

Original entry on oeis.org

1, 1, 3, 4, 9, 13, 25, 33, 56, 76, 122, 170, 264, 360, 538, 707, 1002, 1332, 1853, 2409, 3294

Offset: 0

N. J. A. Sloane, Dec 25 2003

			E.g. a(4) = 9:
4.31.3.22.2.211.21..2..11
.....1....2.....1...1..11
....................1....

N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Cf. A000219, A000123, A018819, A008763, A001970, A089292.

More terms from Wouter Meeussen, Dec 30 2003

A115375 where h[d,d] is a homogeneous symmetric function, s[d,d] is a Schur function indexed by two parts, * represents the Kronecker product and <, > is the standard scalar product on symmetric functions.

Original entry on oeis.org

1, 1, 4, 5, 12, 15, 30, 37, 65, 80, 128, 156, 234, 282, 402, 480, 657, 777, 1030, 1207, 1558, 1811, 2286, 2637, 3267, 3742, 4562, 5192, 6242, 7062, 8388, 9438, 11091, 12417, 14454, 16107, 18592, 20629, 23632, 26117, 29715, 32718, 36996, 40594

Offset: 0

Mike Zabrocki, Jan 21 2006

M. W. Hero and J. F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math., 309 (2010), 6508-6514.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-7,2,8,2,-7,-3,4,1,-1).

Cf. A115376, A082424, A008763, A082437.

Vec((1 - x^2 + x^4) / ((1 - x)^6*(1 + x)^4*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, May 10 2019

G.f.: (1 - x^2 + x^4) / ((1 - x)^6*(1 + x)^4*(1 + x + x^2)).

a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 7*a(n-4) + 2*a(n-5) + 8*a(n-6) + 2*a(n-7) - 7*a(n-8) - 3*a(n-9) + 4*a(n-10) + a(n-11) - a(n-12) for n>11. - Colin Barker, May 10 2019

A115376 where h[d+1,d-1] is a homogeneous symmetric function, s[d,d] is a Schur function indexed by two parts, * represents the Kronecker product and <, > is the standard scalar product on symmetric functions.

Original entry on oeis.org

1, 1, 5, 6, 16, 20, 41, 51, 90, 111, 177, 216, 321, 387, 546, 651, 882, 1041, 1366, 1597, 2042, 2367, 2962, 3407, 4187, 4782, 5787, 6567, 7842, 8847, 10443, 11718, 13692, 15288, 17703, 19677, 22603, 25018, 28532, 31458, 35644, 39158, 44108, 48294

Offset: 2

Mike Zabrocki, Jan 21 2006

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-7,2,8,2,-7,-3,4,1,-1).

Cf. A115375, A082424, A008763, A082437.

Drop[CoefficientList[Series[x^2/((1-x)(1-x^2)^4(1-x^3)),{x,0,50}],x],2]  (* Harvey P. Dale, Aug 24 2011 *)

Vec(x^2 / ((1 - x)^6*(1 + x)^4*(1 + x + x^2)) + O(x^50)) \\ Colin Barker, May 10 2019

G.f.: x^2 / ((1 - x)^6*(1 + x)^4*(1 + x + x^2)).

a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 7*a(n-4) + 2*a(n-5) + 8*a(n-6) + 2*a(n-7) - 7*a(n-8) - 3*a(n-9) + 4*a(n-10) + a(n-11) - a(n-12) for n>11. - Colin Barker, May 10 2019

A251260 Expansion of (1 + 2x + x^2 + x^3) / ((1 - x^2)^2 (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 2, 3, 6, 8, 13, 16, 24, 28, 40, 45, 61, 68, 89, 97, 124, 134, 167, 179, 219, 233, 281, 297, 353, 372, 437, 458, 533, 557, 642, 669, 765, 795, 903, 936, 1056, 1093, 1226, 1266, 1413, 1457, 1618, 1666, 1842, 1894, 2086, 2142, 2350, 2411, 2636, 2701, 2944

Offset: 0

Michael Somos, Mar 20 2015

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 8*x^4 + 13*x^5 + 16*x^6 + 24*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-2,-2,0,1,2,0,-1).

Cf. A008763, A165188, A254594, A254708, A254875.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + 2*x+x^2+x^3)/((1-x^2)^2*(1-x^3)*(1-x^4)))); // G. C. Greubel, Aug 03 2018

a[ n_] := Quotient[ 5 n^3 + If[ OddQ[n], 66 n^2 + 249 n, 57 n^2 + 204 n] + 288, 288];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 6, (u + v < x + w && k == 0) || (u + v > x + w && x + u + v + w == 2 k + 1)}, {x, u, v, w, k}, Integers, 10^9];
LinearRecurrence[{0,2,1,0,-2,-2,0,1,2,0,-1},{1,2,3,6,8,13,16,24,28,40,45},60] (* Harvey P. Dale, Jul 16 2025 *)

{a(n) = (5*n^3 + if( n%2, 66*n^2 + 249*n, 57*n^2 + 204*n) + 288) \ 288};

{a(n) = polcoeff( if( n<0, n = -8-n; -(1 + x + 2*x^2 + x^3), 1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

a(n) = A165188(n+1) + A254708(n-1) = A254594(n-1) + A008763(n+4) for all n in Z.
0 = a(n) - 2*a(n+2) - a(n+3) + 2*a(n+5) + 2*a(n+6) - a(n+8) - 2*a(n+9) + a(n+11) for all n in Z.
a(2*n) = A254875(n) for all n in Z.
G.f.: (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)).

cp's OEIS Frontend

A082424 Coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n), where s(n,n) is the Schur function indexed by two parts of size n, s(2n) is the Schur function corresponding to the trivial representation and * represents the inner or Kronecker product.

Views

Author

Keywords

References

Crossrefs

Programs

Maple

Formula

A082437 Coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n), where s(2n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding two the two row partition and * represents the inner or Kronecker product of symmetric functions.

Views

Author

Keywords

References

Crossrefs

Programs

Maple

Formula

A060962 Expansion of (1+x^2)*(1+x^5)/( Product_{j=1..7} (1-x^j) ).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A089300 Number of planar partitions of n (A000219) that are non-squashing along rows and down columns (cf. A018819).

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A115375 where h[d,d] is a homogeneous symmetric function, s[d,d] is a Schur function indexed by two parts, * represents the Kronecker product and <, > is the standard scalar product on symmetric functions.

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

Formula

A115376 where h[d+1,d-1] is a homogeneous symmetric function, s[d,d] is a Schur function indexed by two parts, * represents the Kronecker product and <, > is the standard scalar product on symmetric functions.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A251260 Expansion of (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A251260 Expansion of (1 + 2x + x^2 + x^3) / ((1 - x^2)^2 (1 - x^3) * (1 - x^4)) in powers of x.