A001993 -id:A001993 - cp's OEIS frontend

A001402 Number of partitions of n into at most 6 parts.

1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 8442, 9192

Offset: 0

N. J. A. Sloane

Also number of partitions of n into parts <= 6: a(n) = A026820(n,6). - Reinhard Zumkeller, Jan 21 2010
Counts unordered closed walks of weight n on a single vertex graph containing 6 loops of weights 1, 2, 3, 4, 5 and 6. - David Neil McGrath, Apr 11 2015
Number of different distributions of n+21 identical balls in 6 boxes as x,y,z,p,q,m where 0Ece Uslu and Esin Becenen, Jan 11 2016
a(n) could be the total number of non-isomorphic geodetic graphs of diameter n>=2 homeomorphic to the Petersen graph. - Carlos Enrique Frasser, May 24 2018

			The number of partitions of 6 into parts less than or equal to 6 is a(6)=11. These are (6)(51)(42)(33)(411)(321)(222)(3111)(2211)(21111)(111111). - _David Neil McGrath_, Apr 11 2015
a(4) = 5, i.e., {1,2,3,4,5,10},{1,2,3,4,6,9},{1,2,3,4,7,8},{1,2,3,5,6,8},{1,2,4,5,6,7} Number of different distributions of 25 identical balls in 6 boxes as x,y,z,p,q,m where 0 < x < y < z < p < q < m. - _Ece Uslu_, Esin Becenen, Jan 11 2016

A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Washington Bomfim, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
G. E. Andrews, Partitions: At the Interface of q-Series and Modular Forms, The Ramanujan Journal 7, 385-400 (2003), Eq.(3.10).
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, American Journal of Mathematics, Vol. 2, No. 1 (Mar., 1879), pp. 71-84 (14 pages).
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. [Annotated scanned copy]
C. E. Frasser and G. N. Vostrov, Geodetic Graphs Homeomorphic to a Given Geodetic Graph, arXiv:1611.01873 [cs.DM], 2016. [p. 27]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 355
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], 2011.
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Essentially same as A026812. Cf. A037145 (first differences), A288341 (partial sums).
a(n) = A008284(n+6, 6), n >= 0.
A194197(n) = a(60*n). - Alois P. Heinz, Aug 23 2011

with(combstruct):ZL7:=[S,{S=Set(Cycle(Z,card<7))}, unlabeled]: seq(count(ZL7,size=n),n=0..50);  # Zerinvary Lajos, Sep 24 2007
a:= n-> (Matrix(21, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 1, 0, 0, -1, 0, -2, 0, 1, 1, 1, 1, 0, -2, 0, -1, 0, 0, 1, 1, -1][i] else 0 fi)^n)[1,1]; seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2008
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=6)},unlabelled]: seq(combstruct[count](B, size=n), n=0..50); # Zerinvary Lajos, Mar 21 2009
## more efficient for large arguments (try with 10^100 or 100^1000):
a:= proc(n) local m, r; m := iquo (n, 60, 'r');
(167 +(2325 +(15400 +(47250 +54000*m +4500*r)*m +3150*r +150*r^2)*m
+[0, 795, 1875, 3030, 4500, 6075, 7995, 10050, 12480, 15075, 18075, 21270, 24900, 28755, 33075, 37650, 42720, 48075, 53955, 60150, 66900, 73995, 81675, 89730, 98400, 107475, 117195, 127350, 138180, 149475, 161475, 173970, 187200, 200955, 215475, 230550, 246420, 262875, 280155, 298050, 316800, 336195, 356475, 377430, 399300, 421875, 445395, 469650, 494880, 520875, 547875, 575670, 604500, 634155, 664875, 696450, 729120, 762675, 797355, 832950][r+1])*m
+[0, 63, 207, 348, 570, 795, 1143, 1482, 1968, 2475, 3135, 3828, 4722, 5643, 6795, 8010, 9468, 11007, 12843, 14760, 17010, 19383, 22107, 24978, 28260, 31695, 35583, 39672, 44238, 49035, 54375, 59958, 66132, 72603, 79695, 87120, 95238, 103707, 112923, 122550, 132960, 143823, 155547, 167748, 180870, 194535, 209163, 224382, 240648, 257535, 275535, 294228, 314082, 334683, 356535, 379170, 403128, 427947, 454143, 481260][r+1])*m/6
+[1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 8442, 9192, 9975, 10829, 11720, 12692, 13702, 14800, 15944, 17180, 18467][r+1] end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 22 2011
A := [1,1,2,3,5,7,11,14,20,26,35,44,58,71,90,110,136,163,199,235,282];
a := proc(n) option remember; if n < 21 then A[n+1] else 1+(a(n-2)+a(n-3)+a(n-4))-(2*a(n-7)+2*a(n-8)+a(n-9))+(a(n-11)+2*a(n-12)+2*a(n-13))-(a(n-16)+a(n-17)+a(n-18))+(a(n-20)) fi end:
seq(a(i),i=0..50); # Peter Luschny, Aug 23 2011
## program using quasi-polynomials; see article by Sills and Zeilberger:
a:= m-> subs (n=m, add ([[n^5/86400 +7*n^4/11520 +77*n^3/6480 +245*n^2/2304 +43981*n/103680 +199577/345600], [-n^2/768 -7*n/256 -581/4608, n^2/768 +7*n/256 +581/4608], [-n/162 -19/324, -n/162 -23/324, n/81 +7/54], [1/32, -1/32, -1/32, 1/32], [1/25, 0, -1/25, -2/25, 2/25], [1/36, -1/36, -1/18, -1/36, 1/36, 1/18]][r][1 +irem (m-1+r, r)], r=1..6)):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 24 2011
## using Andrews-style expressions; see article by Sills and Zeilberger:
a:= n-> 1 +31*n^2/288 +floor(n/4)/16 -floor(n/4 +1/2)/16 +7*n^4/11520 +floor(n/5)/5 +n^5/86400 -(n^2/384 +7*n/128 +581/2304)*n +(n^2/192 +7*n/64 +581/1152) *floor(n/2) -(n/54 +61/324)*n +(n/54 +19/108) *floor((n+1)/3) +(n/27 +7/18) *floor(n/3) +floor(n/6)/18 -floor(n/6 +2/3)/36 +floor(n/6 +1/3)/18 +floor((n+1)/6)/12 +713*n/1800 +77*n^3/6480:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 24 2011

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)*(1 - x^5)*(1 - x^6)), {x, 0, 60} ], x ]
(* Second program: *)
T[n_, k_] := T[n, k] = If[n==0 || k==1, 1, T[n, k-1] + If[k>n, 0, T[n-k, k]]]; a[n_] := T[n, 6]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 12 2017, after Alois P. Heinz's code for A026820 *)
Table[Length[IntegerPartitions[n,6]],{n,0,50}] (* Harvey P. Dale, Jul 30 2025 *)

a(n)=floor((6*n^5+315*n^4+6160*n^3+55125*n^2+(216705+9600*(n%3<1))*n+527500)/518400+(n+1)*(n+20)*(-1)^n/768) \\ Tani Akinari, May 27 2014

a(n)={round((n+11)*((6*n^4+249*n^3+2071*n^2-4931*n+40621)/518400+n\2*(n+10) /192+( (n+1)\3+ n\3*2 )/54))};
vector(60,n,n--; a(n)) \\ Washington Bomfim, Jan 16 2021

a(n) = 1 + (a(n-2) + a(n-3) + a(n-4)) - (2*a(n-7) + 2*a(n-8) + a(n-9)) + (a(n-11) + 2*a(n-12) + 2*a(n-13)) - (a(n-16) + a(n-17) + a(n-18)) + (a(n-20)). - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)). - Alois P. Heinz, Aug 22 2011
a(n) ~ n^5 / 86400. - Charles R Greathouse IV, Aug 23 2011
a(n) = (167 + (2325 + (15400 + (47250 + 54000*m + 4500*r)*m + 3150*r + 150*r^2)*m + X(r))*m + Y(r))*m/6 + Z(r) where m = floor(n/60), r = n mod 60 and X, Y, Z are functions of r (see Maple program below). - Alois P. Heinz, Aug 23 2011
a(n) = floor((2 + 3*(floor(n/3) + floor(-n/3))) * (floor(n/3)+1)/54 + (6*n^5 + 315*n^4 + 6160*n^3 + 55125*n^2 + 219905*n + 485700)/518400 + (n+1)*(n+20)*(-1)^n/768). - Tani Akinari, Aug 05 2013
a(n) = a(n-1) + a(n-2) - a(n-5) - 2*a(n-7) + a(n-9) + a(n-10) + a(n-11) + a(n-12) - 2*a(n-14) - a(n-16) + a(n-19) + a(n-20) - a(n-21). - David Neil McGrath, Apr 11 2015
a(n+6) = a(n) + A001401(n). - Ece Uslu, Esin Becenen, Jan 11 2016
a(n) = round((n+11)*((6*n^4 + 249*n^3 + 2071*n^2 - 4931*n + 40621)/518400 + floor(n/2)*(n+10)/192 + (floor((n+1)/3) + 2*floor(n/3))/54)). - Washington Bomfim, Jan 15 2021

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166, 192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762, 819, 889, 952, 1029, 1099, 1183, 1260, 1352, 1436, 1536, 1628, 1736, 1836, 1953, 2061, 2187, 2304, 2439

Offset: 0

N. J. A. Sloane, Simon Plouffe, Stephen P. Humphries

Number of 2 X 2 square partitions of n.
1/((1-x^2)*(1-x^4)^2*(1-x^6)) is the Molien series for 4-dimensional representation of a certain group of order 192 [Nebe, Rains, Sloane, Chap. 7].
Number of ways of writing n as n = p+q+r+s so that p >= q, p >= r, q >= s, r >= s with p, q, r, s >= 1. That is, we can partition n as
pq
rs
with p >= q, p >= r, q >= s, r >= s.
The coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) is a(n+4), where s(n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding to the two row partition and * represents the inner or Kronecker product of symmetric functions. - Mike Zabrocki, Dec 22 2005
Let F() be the Fibonacci sequence A000045. Let f([x, y, z, w]) = F(x) * F(y) * F(z) * F(w). Let N([x, y, z, w]) = x^2 + y^2 + z^2 + w^2. Let Q(k) = set of all ordered quadruples of integers [x, y, z, w] such that 1 <= x <= y <= z <= w and N([x, y, z, w]) = k. Let P(n) = set of all unordered triples {q1, q2, q3} of elements of some Q(k) such that max(w1, w2, w3) = n and f(q1) + f(q2) = f(q3). Then a(n-1) is the number of elements of P(n). - Michael Somos, Jan 21 2015
Number of partitions of 2n+2 into 4 parts with alternating parity from smallest to largest (or vice versa). - Wesley Ivan Hurt, Jan 19 2021

			a(7) = 4:
41 32 31 22
11 11 21 21
G.f. = x^4 + x^5 + 3*x^6 + 4*x^7 + 7*x^8 + 9*x^9 + 14*x^10 + 17*x^11 + ...
a(5-1) = 1 because P(5) has only one triple {[1,1,1,5], [2,2,2,4], [1,3,3,3]} of elements from Q(28) where f([1,1,1,5]) = 5, f([2,2,2,4]) = 3, f([1,3,3,3]) = 8, and 5 + 3 = 8. - _Michael Somos_, Jan 21 2015
a(6-1) = 1 because P(6) has only one triple {[1,1,2,6], [2,2,3,5], [1,3,4,4]} of elements from Q(42) where f([1,1,2,6]) = 8, f([2,2,3,5]) = 10, f([1,3,4,4]) = 18 and 8 + 10 = 18. - _Michael Somos_, Jan 21 2015
a(7-1) = 3 because P(7) has three triples. The triple {[1,1,1,7], [2,4,4,4], [3,3,3,5]} from Q(52) where f([1,1,1,7]) = 13, f([2,4,4,4]) = 27, f([3,3,3,5]) = 40 and 13 + 27 = 40. The triple {[1,2,2,7], [2,3,3,6], [1,4,4,5]} from Q(58) where f([1,2,2,7]) = 13, f([2,3,3,6]) = 32, f([1,4,4,5]) = 45 and 13 + 32 = 45. The triple {[1,1,3,7], [2,2,4,6], [1,3,5,5]} from Q(60) where f([1,1,3,7]) = 26, f([2,2,4,6]) = 24, f([1,3,5,5]) = 50 and 26 + 24 = 50. - _Michael Somos_, Jan 21 2015

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
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=1).
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

T. D. Noe, Table of n, a(n) for n = 0..1000
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 5.
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2. [MR1219862 (94d:11029)]
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
S. P. Humphries, Home page
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 232
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Michael Somos, In the Elliptic Realm
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

See A266769 for a version without the four leading zeros.
Cf. A001993, A070557, A070558, A070559, A089299, A001970, A089292, A026810, A001400.
First differences of A097701.
Cf. A082424, A082437.

a:=[0,0,0,0,1,1,3,4];; for n in [9..60] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]-2*a[n-4]-a[n-5]+2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Sep 10 2019

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(H);

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( x^4/((1-x)*(1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 10 2019

a:= n-> (Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [1,2,-1,-2,-1,2,1,-1][i] else 0 fi)^n)[1,5]: seq(a(n), n=0..60); # Alois P. Heinz, Jul 31 2008

CoefficientList[Series[x^4/((1-x)*(1-x^2)^2*(1-x^3)), {x,0,60}], x] (* Jean-François Alcover, Mar 30 2011 *)
LinearRecurrence[{1,2,-1,-2,-1,2,1,-1},{0,0,0,0,1,1,3,4},60] (* Harvey P. Dale, Mar 04 2012 *)
a[ n_]:= Quotient[9(n+1)(-1)^n +2n^3 -9n +65, 144]; (* Michael Somos, Jan 21 2015 *)
a[ n_]:= Sign[n] SeriesCoefficient[ x^4/((1-x)(1-x^2)^2(1-x^3)), {x, 0, Abs@n}]; (* Michael Somos, Jan 21 2015 *)

{a(n) = (9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65) \ 144}; /* Michael Somos, Jan 21 2015 */

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,1,2,-1,-2,-1,2,1]^n*[0;0;0;0;1;1;3;4])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

def AA008763_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^4/((1-x)*(1-x^2)^2*(1-x^3))).list()
AA008763_list(60) # G. C. Greubel, Sep 10 2019

Let f4(n) = number of partitions n = p+q+r+s into exactly 4 parts, with p >= q >= r >= s >= 1 (see A026810, A001400) and let g4(n) be the number with q > r (so that g4(n) = f4(n-2)). Then a(n) = f4(n) + g4(n).
a(n) = (1/144)*( 2*n^3 + 9*n*((-1)^n - 1) - 16*((n is 2 mod 3) - (n is 1 mod 3)) ).
a(n) = (1/72)*(n+3)*(n+2)*(n+1)-(1/12)*(n+2)*(n+1)+(5/144)*(n+1)+(1/16)*(n+1)*(-1)^n+(1/16)*(-1)^(n+1)+(7/144)+(2*sqrt(3)/27)*sin(2*Pi*n/3). - Richard Choulet, Nov 27 2008
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8), n>7. - Harvey P. Dale, Mar 04 2012
a(n) = floor((9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65)/144). - Tani Akinari, Nov 06 2012
a(n+1) - a(n) = A008731(n-3). - R. J. Mathar, Aug 06 2013
a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 21 2015
Euler transform of length 3 sequence [1, 2, 1]. - Michael Somos, Jun 26 2017

Entry revised Dec 25 2003

A001996 Number of partitions of n into parts 2, 3, 4, 5, 6, 7.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 17, 23, 26, 33, 37, 47, 52, 64, 72, 86, 96, 115, 127, 149, 166, 192, 212, 245, 269, 307, 338, 382, 419, 472, 515, 576, 629, 699, 760, 843, 913, 1007, 1091, 1197, 1293, 1416, 1525, 1663, 1790, 1945, 2088, 2265, 2426

Offset: 0

N. J. A. Sloane

Also, Molien series for invariants of finite Coxeter group A_6. The Molien series for the finite Coxeter group of type A_k (k >= 1) has G.f. = 1/Prod_{i=2..k+1} (1-x^i). - N. J. A. Sloane, Jan 11 2016

Cayley tabulates the coefficients in the expansion of H = 1 / ((1 - x^2) * (1 - x^4) * ... * (1 - x^14)) with even indices 0, 2, ..., 142.

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
G.f. = 1 + q^2 + q^6 + 2*q^8 + 2*q^10 + 4*q^12 + 4*q^14 + 6*q^16 + ...

A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, American Journal of Mathematics, 2 (1879), pp.71-84. See pp.77-78.
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -2, -2, -1, 0, 2, 2, 2, 2, 0, -1, -2, -2, -1, 0, 0, 1, 1, 1, 0, -1).

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

nn = 102; t = CoefficientList[Series[1/((1 - x^4)*(1 - x^6)*(1 - x^8)*(1 - x^10)*(1 - x^12)*(1 - x^14)), {x, 0, nn}], x]; t = Take[t, {1, nn, 2}]

G.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).

Euler transform of length 7 sequence [ 0, 1, 1, 1, 1, 1, 1]. - Michael Somos, Apr 23 2014

More terms from James Sellers, Feb 09 2000

A052282 Number of 3 X 3 stochastic matrices under row and column permutations.

Original entry on oeis.org

1, 1, 3, 5, 9, 13, 22, 30, 45, 61, 85, 111, 149, 189, 244, 304, 381, 465, 571, 685, 825, 977, 1158, 1354, 1585, 1833, 2121, 2431, 2785, 3165, 3596, 4056, 4573, 5125, 5739, 6393, 7117, 7885, 8730, 9626, 10605, 11641, 12769, 13959, 15249, 16609, 18076, 19620

Offset: 0

Vladeta Jovovic, Feb 06 2000

nonn

Unreduced numerators in convergent to log(2) = lim[n->inf, a(n)/A000670(n+1)].

			There are 5 nonisomorphic 3 X 3 matrices with row and column sums 3:
[0 0 3] [0 0 3] [0 1 2] [0 1 2] [1 1 1]
[0 3 0] [1 2 0] [1 1 1] [1 2 0] [1 1 1]
[3 0 0] [2 1 0] [2 1 0] [2 0 1] [1 1 1]

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Row n=3 of A333733.

Cf. A002817, A052280, A052281. Different from A001993.

a:= n -> (Matrix([[1, 0, 0, 1, 1, 3, 5, 9, 13]]). Matrix(9, (i,j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -3, -1, 1, 3, -1, -2, 1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2008

LinearRecurrence[{2,1,-3,-1,1,3,-1,-2,1},{1,1,3,5,9,13,22,30,45},50] (* Harvey P. Dale, Mar 10 2018 *)

G.f.: (x^6-x^5+x^3-x+1)/((1-x)^5*(1+x)^2*(1+x+x^2)). - Ralf Stephan and Vladeta Jovovic, May 07 2004

cp's OEIS Frontend

A001402 Number of partitions of n into at most 6 parts.

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A008763 Expansion of g.f.: x^4/((1-x)*(1-x^2)^2*(1-x^3)).

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A001996 Number of partitions of n into parts 2, 3, 4, 5, 6, 7.

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A052282 Number of 3 X 3 stochastic matrices under row and column permutations.

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A070557 Number of two-rowed partitions of length 4.

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A070558 Number of two-rowed partitions of length 5.

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A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

A001994 Expansion of 1/((1-x^2)(1-x^4)^2(1-x^6)(1-x^8)(1-x^10)) (even powers only).