A001400 -id:A001400 - cp's OEIS frontend

A001971 Nearest integer to n^2/8.

0, 0, 1, 1, 2, 3, 5, 6, 8, 10, 13, 15, 18, 21, 25, 28, 32, 36, 41, 45, 50, 55, 61, 66, 72, 78, 85, 91, 98, 105, 113, 120, 128, 136, 145, 153, 162, 171, 181, 190, 200, 210, 221, 231, 242, 253, 265, 276, 288, 300, 313, 325, 338, 351, 365, 378, 392, 406, 421, 435, 450

Offset: 0

N. J. A. Sloane

Restricted partitions.
a(0) = a(1) = 0; a(n) are the partitions of floor((3*n+3)/2) with 3 distinct numbers of the set {1, ..., n}; partitions of floor((3*n+3)/2)-C and ceiling((3*n+3)/2)+C have equal numbers. - Paul Weisenhorn, Jun 05 2009, corrected by M. F. Hasler, Jun 16 2022
Odd-indexed terms are the triangular numbers, even-indexed terms are the midpoint (rounded up where necessary) of the surrounding odd-indexed terms. - Carl R. White, Aug 12 2010
a(n+2) is the number of points one can surround with n stones in Go (including the points under the stones). - Thomas Dybdahl Ahle, May 11 2014
Corollary of above: a(n) is the number of points one can surround with n+2 stones in Go (excluding the points under the stones). - Juhani Heino, Aug 29 2015
From Washington Bomfim, Jan 13 2021: (Start)
For n >= 4, a(n) = A026810(n+2) - A026810(n-4).
Let \n,m\ be the number of partitions of n into m non-distinct parts.
For n >= 1, \n,4\ = round((n-2)^2/8).
For n >= 6, \n,4\ = A026810(n) - A026810(n-6).
(End)

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
M. Jeger, Einfuehrung in die Kombinatorik, Klett, 1975, Bd.2, pages 110 ff. [Paul Weisenhorn, Jun 05 2009]
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
G. Almkvist, Invariants, mostly old ones, Pacific J. Math. 86 (1980), no. 1, 1-13. MR0586866 (81j:14029)
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Shalosh B. Ekhad and Doron Zeilberger, In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?, arXiv:1901.08172 [math.CO], 2019.
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.
D. Vainsencher and A. M. Bruckstein, On isoperimetrically optimal polyforms, Theoretical Computer Science 406.1-2, 2008, pp. 146-159.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

The 4th diagonal of A061857?
Kind of an inverse of A261491 (regarding Go).
Cf. A026810 (partitions with greatest part 4), A001400 (partitions in at most 4 parts), A000217 (a(2n+1): triangular numbers n(n+1)/2), A000982 (a(2n): round(n^2/2)).

a001971 = floor . (+ 0.5) . (/ 8) . fromIntegral . (^ 2)
-- Reinhard Zumkeller, May 08 2012

[Round(n^2/8): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011

A001971:=-(1-z+z**2)/((z+1)*(z**2+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation [Note that this "generating function" is Sum_{n >= 0} a(n+2)*z^n, not a(n)*z^n. - M. F. Hasler, Jun 16 2022]

LinearRecurrence[{2,-1,0,1,-2,1},{0,0,1,1,2,3},70] (* Harvey P. Dale, Jan 30 2014 *)

PARI
```
{a(n) = round(n^2 / 8)};
```

apply( {A001971(n)=n^2\/8}, [0..99]) \\ M. F. Hasler, Jun 16 2022

The listed terms through a(20)=50 satisfy a(n+2) = a(n-2) + n. - John W. Layman, Dec 16 1999
G.f.: x^2 * (1 - x + x^2) / (1 - 2*x + x^2 - x^4 + 2*x^5 - x^6) = x^2 * (1 - x^6) / ((1 - x) * (1 - x^2) * (1 - x^3) * (1 - x^4)). - Michael Somos, Feb 07 2004
a(n) = floor((n^2+4)/8). - Paul Weisenhorn, Jun 05 2009
a(2*n+1) = A000217(n), a(2*n) = floor((A000217(n-1)+A000217(n)+1)/2). - Carl R. White, Aug 12 2010
From Michael Somos, Aug 29 2015: (Start)
Euler transform of length 6 sequence [ 1, 1, 1, 1, 0, -1].
a(n) = a(-n) for all n in Z. (End)
a(2n) = A000982(n). - M. F. Hasler, Jun 16 2022
Sum_{n>=2} 1/a(n) = 2 + Pi^2/12 + tanh(Pi/2)*Pi/2. - Amiram Eldar, Jul 02 2023

Edited Feb 08 2004

A008804 Expansion of 1/((1-x)^2(1-x^2)(1-x^4)).

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 20, 26, 35, 44, 56, 68, 84, 100, 120, 140, 165, 190, 220, 250, 286, 322, 364, 406, 455, 504, 560, 616, 680, 744, 816, 888, 969, 1050, 1140, 1230, 1330, 1430, 1540, 1650, 1771, 1892, 2024, 2156, 2300, 2444, 2600, 2756, 2925, 3094, 3276, 3458

Offset: 0

N. J. A. Sloane

b(n)=a(n-3) is the number of asymmetric nonnegative integer 2 X 2 matrices with sum of elements equal to n, under action of dihedral group D_4(b(0)=b(1)=b(2)=0). G.f. for b(n) is x^3/((1-x)^2*(1-x^2)*(1-x^4)). - Vladeta Jovovic, May 07 2000
If the offset is changed to 5, this is the 2nd Witt transform of A004526 [Moree]. - R. J. Mathar, Nov 08 2008
a(n) is the number of partitions of 2*n into powers of 2 less than or equal to 2^3. First differs from A000123 at n=8. - Alois P. Heinz, Apr 02 2012
a(n) is the number of bracelets with 4 black beads and n+3 white beads which have no reflection symmetry. For n=1 we have for example 2 such bracelets with 4 black beads and 4 white beads: BBBWBWWW and BBWBWBWW. - Herbert Kociemba, Nov 27 2016
a(n) is the also number of aperiodic bracelets with 4 black beads and n+3 white beads which have no reflection symmetry. This is equivalent to saying that a(n) is the (n+7)th element of the DHK[4] (bracelet, identity, unlabeled, 4 parts) transform of 1, 1, 1, ... (see Bower's link about transforms). Thus, for n >= 1 , a(n) = (DHK[4] c){n+7}, where c = (1 : n >= 1). This is because every bracelet with 4 black beads and n+3 white beads which has no reflection symmetry must also be aperiodic. This statement is not true anymore if we have k black beads where k is even >= 6. - _Petros Hadjicostas, Feb 24 2019

			G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 14*x^5 + 20*x^6 + 26*x^7 + 35*x^8 + ...
There are 10 asymmetric nonnegative integer 2 X 2 matrices with sum of elements equal to 7 under action of D_4:
[0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1]
[1 6] [2 5] [3 4] [2 4] [3 3] [4 2] [5 1] [3 2] [4 1] [2 3]

T. D. Noe, Table of n, a(n) for n = 0..1000
C. G. Bower, Transforms (2)
Petros Hadjicostas, The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry, 2019.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 197
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From _R. J. Mathar_, Nov 08 2008]
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Cf. A000123, A001400, A002620, A004526, A005232, A014557, A032246, A032248, A053307.

Column k=3 of A181322. Column k = 4 of A180472 (but with different offset).

a:=[1,2,4,6,10,14,20,26];; for n in [9..60] do a[n]:=2*a[n-1] -2*a[n-3]+2*a[n-4]-2*a[n-5]+2*a[n-7]-a[n-8]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)^2*(1-x^2)*(1-x^4)) )); // G. C. Greubel, Sep 12 2019

seq(coeff(series(1/((1-x)^2*(1-x^2)*(1-x^4)), x, n+1), x, n), n = 0..60); # G. C. Greubel, Sep 12 2019

LinearRecurrence[{2,0,-2,2,-2,0,2,-1}, {1,2,4,6,10,14,20,26}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
gf[x_,k_]:=x^k/2 (1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])-(1+x)/(1-x^2)^Floor[k/2+1]); CoefficientList[Series[gf[x,4]/x^7,{x,0,60}],x] (* Herbert Kociemba, Nov 27 2016 *)
Table[(84 +12*(-1)^n +85*n +3*(-1)^n*n +24*n^2 +2*n^3 +12*Sin[n Pi/2])/96, {n,0,60}] (* Eric W. Weisstein, Oct 12 2017 *)
CoefficientList[Series[1/((1-x)^4*(1+x)^2*(1+x^2)), {x,0,60}], x] (* Eric W. Weisstein, Oct 12 2017 *)

a(n)=(84+12*(-1)^n+6*I*((-I)^n-I^n)+(85+3*(-1)^n)*n+24*n^2 +2*n^3)/96 \\ Jaume Oliver Lafont, Sep 20 2009

{a(n) = my(s = 1); if( n<-7, n = -8 - n; s = -1); if( n<0, 0, s * polcoeff( 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^4)) + x * O(x^n), n))}; /* Michael Somos, Feb 02 2011 */

def A008804_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)^2*(1-x^2)*(1-x^4))).list()
A008804_list(60) # G. C. Greubel, Sep 12 2019

For a formula for a(n) see A014557.
a(n) = (84 +85*n +24*n^2 +2*n^3 +12*A056594(n+3) +3*(-1)^n*(n+4))/96. - R. J. Mathar, Nov 08 2008
a(n) = 2*(Sum_{k=0..floor(n/2)} A002620(k+2)) - A002620(n/2+2)*(1+(-1)^n)/2. - Paul Barry, Mar 05 2009
G.f.: 1/((1-x)^4*(1+x)^2*(1+x^2)). - Jaume Oliver Lafont, Sep 20 2009
Euler transform of length 4 sequence [2, 1, 0, 1]. - Michael Somos, Feb 05 2011
a(n) = -a(-8 - n) for all n in Z. - Michael Somos, Feb 05 2011
From Herbert Kociemba, Nov 27 2016: (Start)
More generally gf(k) is the g.f. for the number of bracelets without reflection symmetry with k black beads and n-k white beads.
gf(k): x^k/2 * ( (1/k)*Sum_{n|k} phi(n)/(1 - x^n)^(k/n) - (1 + x)/(1 -x^2)^floor(k/2 + 1) ). The g.f. here is gf(4)/x^7 because of the different offset. (End)
E.g.f.: ((48 + 54*x + 15*x^2 + x^3)*cosh(x) + 6*sin(x) + (36 + 57*x + 15*x^2 + x^3)*sinh(x))/48. - Stefano Spezia, May 15 2023
a(n) = A001400(n) + A001400(n-1) + A001400(n-2). - David García Herrero, Aug 26 2024
a(n) = floor((2*n^3 + 24*n^2 + n*(85+3*(-1)^n) + 96) / 96). - Hoang Xuan Thanh, May 24 2025

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

A060016 Triangle T(n,k) = number of partitions of n into k distinct parts, 1 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, 1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

Also number of partitions of n-k(k+1)/2 into at most k parts (not necessarily distinct).

A025147(n) = Sum_{k=2..floor((n+2)/2)} a(n-k+1, k-1). - Reinhard Zumkeller, Nov 04 2007

			Triangle starts
[ 1]  1,
[ 2]  1, 0,
[ 3]  1, 1, 0,
[ 4]  1, 1, 0, 0,
[ 5]  1, 2, 0, 0, 0,
[ 6]  1, 2, 1, 0, 0, 0,
[ 7]  1, 3, 1, 0, 0, 0, 0,
[ 8]  1, 3, 2, 0, 0, 0, 0, 0,
[ 9]  1, 4, 3, 0, 0, 0, 0, 0, 0,
[10]  1, 4, 4, 1, 0, 0, 0, 0, 0, 0,
[11]  1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0,
[12]  1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0,
[13]  1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0,
[14]  1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, ...
T(8,3)=2 since 8 can be written in 2 ways as the sum of 3 distinct positive integers: 5+2+1 and 4+3+1.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 219.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.

Alois P. Heinz, Rows n = 1..141, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Columns (offset) include A057427, A004526, A001399, A001400, A001401, etc. Cf. A000009 (row sums), A008289 (without zeros), A030699 (row maximum), A008284 (partition triangle including duplications).

See A008289 for another version.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
T:= proc(n) local l; l:= subsop(1=NULL, b(n, n));
      l[], 0$(n-nops(l))
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Dec 12 2012

Flatten[Table[Length[Select[IntegerPartitions[n,{k}],Max[Transpose[ Tally[#]][[2]]]==1&]],{n,20},{k,n}]] (* Harvey P. Dale, Feb 27 2012 *)
T[, 1] = 1; T[n, k_] /; 1, ] = 0; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 26 2015 *)

N=16;  q='q+O('q^N);
gf=sum(n=0,N, z^n * q^((n^2+n)/2) / prod(k=1,n, 1-q^k ) );
/* print triangle: */
gf -= 1; /* remove row zero */
P=Pol(gf,'q);
{ for (n=1,N-1,
    p = Pol(polcoeff(P, n),'z);
    p += 'z^(n+1);  /* preserve trailing zeros */
    v = Vec(polrecip(p));
    v = vector(n,k,v[k]); /* trim to size n */
    print(v);
); }
/* Joerg Arndt, Oct 20 2012 */

T(n, k) = T(n-k, k) + T(n-k, k-1) [with T(n, 0)=1 if n=0 and 0 otherwise].

G.f.: Sum_{n>=0} z^n * q^((n^2+n)/2) / Product_{k=1..n} (1-q^k), rows by powers of q, columns by powers of z; includes row 0 (drop term for n=0 for this triangle, see PARI code); setting z=1 gives g.f. for A000009; cf. to g.f. for A072574. - Joerg Arndt, Oct 20 2012

More terms, recurrence, etc. from Henry Bottomley, Mar 26 2001

A050913 Pure 2-complexes on an infinite set of nodes with n multiple 2-simplexes. Also n-rowed binary matrices with all row sums 3, up to row and column permutation.

Original entry on oeis.org

1, 1, 4, 16, 93, 652, 6369, 79568, 1256425, 24058631, 543204998, 14138916124, 417362929209, 13798729189578, 505990335048034, 20415765544541866, 900364519682003919, 43155049922002494115, 2236988329443856718604, 124862936181977439454012, 7476052709321753156375756, 478506183522725779096476581, 32638841238874891261354722405, 2365895836144423508306322639848, 181785988254681334224483607437510, 14771116583797935886529061991645404, 1266545494725474774697216198539818982

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation

Row n=3 of A331461.

Cf. A050911, A050912, A001400, A014395, A082789, A082790, A058790.

More terms from T. Forbes (anthony.d.forbes(AT)googlemail.com), May 24 2003

A002621 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)*(1-x^4)).

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 27, 38, 53, 71, 94, 121, 155, 194, 241, 295, 359, 431, 515, 609, 717, 837, 973, 1123, 1292, 1477, 1683, 1908, 2157, 2427, 2724, 3045, 3396, 3774, 4185, 4626, 5104, 5615, 6166, 6754, 7386, 8058, 8778, 9542, 10358, 11222, 12142, 13114

Offset: 0

N. J. A. Sloane

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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 199
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
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1).

Partial sums of A001400. Column 4 of A092905.

A002621:=-1/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**5; # Simon Plouffe in his 1992 dissertation
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+2), right=Set(U, card=1)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=4..51) ; # Zerinvary Lajos, Feb 07 2008
A057077 := proc(n) (-1)^floor(n/2) ; end proc:
A061347 := proc(n) op(1+(n mod 3),[1,1,-2]) ; end proc:
A002621 := proc(n) 83/288*n^2+55/64*n+2815/3456+11/288*n^3+1/576*n^4+11/128*(-1)^n+1/64*(-1)^n*n; %+ A057077(n)/16 +A061347(n)/27; end proc:
seq(A002621(n),n=0..10) ; # R. J. Mathar, Mar 15 2011

CoefficientList[Series[1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^4)),{x,0,60}],x] (* Stefan Steinerberger, Jun 10 2007 *)
LinearRecurrence[{2, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1}, {1, 2, 4, 7, 12, 18, 27, 38, 53, 71, 94}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)

a(n)=(n+1)*(9*(-1)^n+n^3+21*n^2+145*n+350)\/576 \\ Charles R Greathouse IV, May 23 2013

a(n) = +2*a(n-1) - a(n-3) - 2*a(n-5) + 2*a(n-6) + a(n-8) - 2*a(n-10) + a(n-11).
a(n) = 83*n^2/288 +55*n/64 +2815/3456 +11*n^3/288 +n^4/576 +11*(-1)^n/128 +(-1)^n*n/64 + A057077(n)/16 +A061347(n)/27. - R. J. Mathar, Mar 15 2011
a(n)=floor((n+1)*(9*(-1)^n + n^3 + 21*n^2 + 145*n + 350)/576 + 1/2). - Tani Akinari, Nov 10 2012

A000710 Number of partitions of n, with two kinds of 1, 2, 3 and 4.

Original entry on oeis.org

1, 2, 5, 10, 20, 35, 62, 102, 167, 262, 407, 614, 919, 1345, 1952, 2788, 3950, 5524, 7671, 10540, 14388, 19470, 26190, 34968, 46439, 61275, 80455, 105047, 136541, 176593, 227460, 291673, 372605, 474085, 601105, 759380, 956249, 1200143, 1501749, 1873407

Offset: 0

N. J. A. Sloane

Also number of partitions of 2*n+4 with exactly 4 odd parts. - Vladeta Jovovic, Jan 12 2005
Convolution of A000041 and A001400. - Vaclav Kotesovec, Aug 18 2015
Also the sum of binomial (D(p), 4) over partitions p of n+10, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

			a(2) = 5 because we have 2, 2', 1+1, 1+1', 1'+1'.

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A000712.
Cf. A000070, A008951, A000097, A000098.
Fifth column of Riordan triangle A008951 and of triangle A103923.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<5,2,1)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[If[#<5, 2, 1]&]; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3)(1-x^4))*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@4], {n,0,39}] (* Robert Price, Jul 28 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[, ] = 0;
a[n_] := T[n + 10, 4];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Euler transform of 2 2 2 2 1 1 1...
G.f.: 1/((1-x)(1-x^2)(1-x^3)(1-x^4)*Product_{k>=1} (1-x^k)).
a(n) = Sum_{j=0..floor(n/4)} A000098(n-4*j), n >= 0.
a(n) ~ sqrt(3)*n * exp(Pi*sqrt(2*n/3)) / (8*Pi^4). - Vaclav Kotesovec, Aug 18 2015

Edited by Emeric Deutsch, Mar 22 2005

A218320 Number of ways to write n as n = abc*d with 1 <= a <= b <= c <= d <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 6, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 9, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 11, 2

Offset: 1

Michel Lagneau, Oct 25 2012

nonn

Starts the same as, but is different from A001055. First values of n such that a(n) differs from A001055(n) are 32, 48, 64, 72, 80, ... .

The value of a is the same for all numbers n with the same prime signature. For prime p we have a(p^n) = A001400(n), the number of partitions of n into at most 4 parts. - Alois P. Heinz, Nov 03 2012

			a(12) = 4 because we can write 12 = 1*1*1*12 = 1*1*2*6 = 1*1*3*4 = 1*2*2*3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001055, A034836.

for n from 1 to 90 do:t1:=0: for a from 1 to n do: for b from a to n do :for c from b to n do : for d from c to n do :if a*b*c*d = n then t1:=t1+1: else fi: od: od: od: od:printf(`%d, `,t1):od:
# second Maple program
with(numtheory):
b:= proc(n, i, t) option remember;
      `if`(n=1, 1, `if`(t=1, `if`(n<=i, 1, 0),
       add(b(n/d, d, t-1), d=select(x->x<=i, divisors(n)))))
    end:
a:= proc(n) local l, m;
      l:= sort(ifactors(n)[2], (x, y)-> x[2]>y[2]);
      m:= mul(ithprime(i)^l[i][2], i=1..nops(l));
      b(m, m, 4)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2012

b[n_, i_, t_] := b[n, i, t] = If[n==1, 1, If[t==1, If[n <= i, 1, 0], Sum[b[n/d, d, t-1], {d, Select[Divisors[n], # <= i&]}]]];
a[n_] := (l = Sort[FactorInteger[n], #1[[2]] > #2[[2]]&]; m = Times @@ Power @@@ l; b[m, m, 4]);
Array[a, 100] (* Jean-François Alcover, Mar 22 2017, after Alois P. Heinz *)

A000711 Number of partitions of n, with three kinds of 1,2,3 and 4 and two kinds of 5,6,7,...

Original entry on oeis.org

1, 3, 9, 22, 51, 107, 217, 416, 775, 1393, 2446, 4185, 7028, 11569, 18749, 29908, 47083, 73157, 112396, 170783, 256972, 383003, 565961, 829410, 1206282, 1741592, 2497425, 3557957, 5037936, 7091711, 9927583, 13823626, 19151731, 26404879, 36236988, 49509149

Offset: 0

N. J. A. Sloane

nonn

Convolution of A000712 and A001400. - Vaclav Kotesovec, Aug 18 2015

			a(2)=9 because we have 2, 2', 2", 1+1, 1'+1', 1"+1", 1+1', 1+1", 1'+1".

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
N. J. A. Sloane, Transforms

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<5,3,2)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

nn=31;CoefficientList[Series[1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/Product[(1-x^i)^2,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 28 2013 *)

EULER transform of 3, 3, 3, 3, 2, 2, 2, 2, ...
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*Product_{k>=1} (1 - x^k)^2).
a(n) ~ exp(2*Pi*sqrt(n/3)) * 3^(1/4) * n^(3/4) / (32*Pi^4). - Vaclav Kotesovec, Aug 18 2015

Extended with formula from Christian G. Bower, Apr 15 1998

Edited by Emeric Deutsch, Mar 22 2005

A061924 Number of combinations in card games with 4 suits and 4 players.

Original entry on oeis.org

1, 4, 56, 1320, 43680, 1860480, 96909120, 5967561600, 424097856000, 34162713446400, 3075990524006400, 306135476264217600, 33371339479827148800, 3954242643911239680000, 506046613478104258560000, 69560546966425756200960000, 10221346459144248675287040000

Offset: 0

Frank Ellermann, May 16 2001

			a(13) = 52*51*50*49*48*47*46*45*44*43*42*41*40 (Whist, Bridge).

Harry J. Smith, Table of n, a(n) for n=0..100
Frank Ellermann, Illustration for A001400, A061924

Cf. A001400.

Table[(4n)!/(4n-n)!,{n,0,20}] (* Harvey P. Dale, Aug 20 2012 *)

{ for (n=0, 100, write("b061924.txt", n, " ", (4*n)! / (4*n - n)!) ) } \\ Harry J. Smith, Jul 29 2009

from sympy import ff
def A061924(n): return ff(n<<2,n) # Chai Wah Wu, Sep 01 2023

a(n) = (4*n)! / (4*n-n)!

E.g.f. in Maple notation: hypergeom([1/2, 1/4, 3/4], [1/3, 2/3], 256*x/27). - Karol A. Penson, Oct 18 2001

More terms from Harvey P. Dale, Aug 20 2012

cp's OEIS Frontend

A001971 Nearest integer to n^2/8.

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A008804 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^4)).

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

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A060016 Triangle T(n,k) = number of partitions of n into k distinct parts, 1 <= k <= n.

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A050913 Pure 2-complexes on an infinite set of nodes with n multiple 2-simplexes. Also n-rowed binary matrices with all row sums 3, up to row and column permutation.

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A002621 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*(1-x^4)).

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A008804 Expansion of 1/((1-x)^2(1-x^2)(1-x^4)).

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

A002621 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)*(1-x^4)).

A218320 Number of ways to write n as n = abc*d with 1 <= a <= b <= c <= d <= n.