A001971 -id:A001971 - cp's OEIS frontend

A000601 Expansion of 1/((1-x)^2(1-x^2)(1-x^3)).

1, 2, 4, 7, 11, 16, 23, 31, 41, 53, 67, 83, 102, 123, 147, 174, 204, 237, 274, 314, 358, 406, 458, 514, 575, 640, 710, 785, 865, 950, 1041, 1137, 1239, 1347, 1461, 1581, 1708, 1841, 1981, 2128, 2282, 2443, 2612, 2788, 2972, 3164, 3364, 3572, 3789, 4014

Offset: 0

N. J. A. Sloane

Molien series for 4-dimensional representation of S_3 [Nebe, Rains, Sloane, Chap. 7].
From Thomas Wieder, Feb 11 2007: (Start)
If P(i,k) denotes the number of integer partitions of i into k parts and if k=3, then a(n) = Sum_{i=k..n+2} P(i,k). See also A002620 = Quarter-squares, this sequence follows for k=2 as pointed out by Rick L. Shepherd, Feb 27 2004.
For example, a(n=6)=16 because there are 16 integer partitions of n=3,4,...,n+2=8 with k=3 parts:
[[1, 1, 1]],
[[2, 1, 1]],
[[3, 1, 1], [2, 2, 1]]
[[4, 1, 1], [3, 2, 1], [2, 2, 2]],
[[5, 1, 1], [4, 2, 1], [3, 3, 1], [3, 2, 2]],
[[6, 1, 1], [5, 2, 1], [4, 3, 1], [4, 2, 2], [3, 3, 2]]. (End)
Let P(i,k) be the number of integer partitions of n into k parts. Then if k=3 we have a(n) = Sum_{i=k..n} P(i,k=3). - Thomas Wieder, Feb 20 2007
Number of equivalence classes of 3 X n binary matrices when one can permute rows, permute columns and complement columns. - Max Alekseyev, Feb 05 2010
Convolution of the sequences whose n-th terms are given by 1+[n/2] and 1+[n/3], where []=floor. - Clark Kimberling, May 28 2012
Number of partitions of n into two sorts of 1, and one sort each of 2 and 3. - Joerg Arndt, May 05 2014
a(n-3) is the number of partitions mu of 2n of length 4 such that mu has an even number of even entries and the transpose of mu has an even number of even entries (see below example). - John M. Campbell, Feb 03 2016
Number of partitions of 2n+8 into 4 parts such that the sum of the smallest two parts and the sum of the largest two parts are both odd. Also, number of partitions of 2n+4 into 4 parts such that the sum of the smallest two parts and the sum of the largest two parts are both even. - Wesley Ivan Hurt, Jan 19 2021

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 23*x^6 + 31*x^7 + ...
From _John M. Campbell_, Feb 03 2016:  (Start)
For example, letting n=6, there are a(n-3)=a(3)=7 partitions mu of 12 of length 4 such mu has an even number of even entries and the transpose of mu has an even number of even entries: (8,2,1,1), (6,4,1,1), (6,3,2,1), (6,2,2,2), (4,4,3,1), (4,4,2,2), (4,3,3,2). For example, the partition
   oooooo
   ooo
   oo
   o
has 2 even entries and the transpose
   oooo
   ooo
   oo
   o
   o
   o
has an even number of even entries. (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.
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 Vincenzo Librandi)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
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]
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
Florent de Dinechin, Matei Istoan, Guillaume Sergent, Kinga Illyes, Bogdan Popa and Nicolas Brunie, Arithmetic around the bit heap, HAL Id: ensl-00738412, 2012. - From _N. J. A. Sloane_, Dec 31 2012
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]
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649 - _Max Alekseyev_, Feb 05 2010
H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085.
H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 196
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
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,-1,0,2,-1)
Index entries for two-way infinite sequences

Cf. A002620, A006148, A006383.

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; U:=MatrixGroup<4,K|q2,h>; G:=MatrixGroup<4,K|q1,q2,h>; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(U);

A000601:=1/(z+1)/(z**2+z+1)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+1), right=Set(U, card=1)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=3..52) ; # Zerinvary Lajos, Feb 07 2008

CoefficientList[Series[1/((1-x)^2*(1-x^2)*(1-x^3)), {x, 0, 49}], x] (* Jean-François Alcover, Jul 20 2011 *)
LinearRecurrence[{2,0,-1,-1,0,2,-1},{1,2,4,7,11,16,23},50] (* Harvey P. Dale, Mar 17 2013 *)
a[ n_] := Quotient[ 2 n^3 + 21 n^2 + 66 n, 72] + 1; (* Michael Somos, May 28 2014 *)

Vec(1/((1-x)^2*(1-x^2)*(1-x^3))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

{a(n) = (2*n^3 + 21*n^2 + 66*n) \ 72 + 1}; /* Michael Somos, May 28 2014 */

a(n) = n^3/36 +7*n^2/24 +11*n/12 +119/144 +(-1)^n/16 + A057078(n)/9. - R. J. Mathar, Mar 14 2011
a(0)=1, a(1)=2, a(2)=4, a(3)=7, a(4)=11, a(5)=16, a(6)=23, a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7). - Harvey P. Dale, Mar 17 2013
It appears that a(n) = ((4*n^3+42*n^2+140*n+102+21*(1+(-1)^n))/8-6*floor((2*n+5+3*(-1)^n)/12))/18. - Luce ETIENNE, May 05 2014
Euler transform of length 3 sequence [ 2, 1, 1]. - Michael Somos, May 28 2014
a(-7 - n) = -a(n). - Michael Somos, May 28 2014

More terms from James Sellers, Feb 06 2000

A059169 Number of partitions of n into 3 parts which form the sides of a nondegenerate isosceles triangle.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 7, 8, 8, 9, 8, 9, 9, 10, 9, 10, 10, 11, 10, 11, 11, 12, 11, 12, 12, 13, 12, 13, 13, 14, 13, 14, 14, 15, 14, 15, 15, 16, 15, 16, 16, 17, 16, 17, 17, 18, 17, 18, 18, 19, 18, 19, 19, 20, 19, 20, 20

Offset: 1

Floor van Lamoen, Jan 13 2001

Also number of 0's in n-th row of triangle in A071026. - Hans Havermann, May 26 2002
Exponent of 2 in factorization of A030436(n-1) and A026655(n-1). First differences of A001971. - Ralf Stephan, Mar 21 2004
Conjecture: this is 0 followed by A026922. - R. J. Mathar, Oct 05 2008 [See the g.f. given there by Michael Somos and the one given below for the proof. - Wolfdieter Lang, May 10 2017]
a(n+1) is for n >= 0 the number of integers k in the left-sided open interval ((n+1)/4, floor(n/2)]. This is needed for the number of zeros of Chebyshev S polynomials in the open interval (-sqrt(2), sqrt(2)) given in A285869. - Wolfdieter Lang, May 10 2017

			Consider the number 13. The following partitions give a nondegenerate triangle: 4 4 5; 3 5 5; 1 6 6; 2 5 6; 3 4 6. Since the first three partitions represent isosceles triangles, we have A059169(13) = 3.
G.f. = x^3 + x^5 + x^6 + 2*x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 2*x^12 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Essentially the same as A008624.

Cf. A178804.

a059169 n = a059169_list !! (n-1)
a059169_list = map abs $ zipWith (-) (tail a178804_list) a178804_list
-- Reinhard Zumkeller, Nov 15 2014

[Floor((n-1)/2) - Floor(n/4): n in [1..80]]; // G. C. Greubel, Mar 08 2018

a[1] := 0: a[2] := 0: a[3] := 1: a[4] := 0: a[5] := 1: for n from 6 to 300 do a[n] := a[n-1] + a[n-4] - a[n-5]: end do: seq(a[n], n=1..82);
a := n -> A005044(n) - A005044(n-6): A005044 := n-> floor((1/48)*(n^2 + 3*n + 21 + (-1)^(n-1)*3*n)): seq(a(n), n = 1..82); # Johannes W. Meijer, Oct 10 2013

CoefficientList[Series[x^2 (1 - x + x^2)/(1 - x - x^4 + x^5), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 15 2013 *)
LinearRecurrence[{1,0,0,1,-1},{0,0,1,0,1},100] (* Harvey P. Dale, Feb 09 2015 *)
a[ n_] := Quotient[ n - 1, 2] - Quotient[ n, 4]; (* Michael Somos, May 05 2015 *)

{a(n) = (n - 1) \ 2 - (n \ 4)}; /* Michael Somos, Oct 14 2008 */

{a(n) = if( n<1, -a(3 - n), polcoeff( x^3 * (1 - x + x^2) / (1 - x - x^4 + x^5) + x * O(x^n), n))}; /* Michael Somos, Oct 14 2008 */

a(2*n + 2) = a(2*n - 1) = A004526(n).
a(n) = A005044(n) - A005044(n-6).
From Vladeta Jovovic, Dec 29 2001: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: x^3*(1 - x + x^2)/(1 - x - x^4 + x^5). (End)
The g.f. can also be written as x^3 * (1 + x^3) / ((1 - x^2) * (1 - x^4)). - Michael Somos, May 05 2015
Euler transform of length 6 sequence [0, 1, 1, 1, 0, -1]. - Michael Somos, Oct 14 2008
a(n) = -a(3 - n) for all n in Z. - Michael Somos. Oct 14 2008
a(n) = abs(floor((n-1)*(-1)^n/4)). - Wesley Ivan Hurt, Oct 22 2013
a(n) = abs(A178804(n+1) - A178804(n)). - Reinhard Zumkeller, Nov 15 2014
a(n) = floor(n/2) - floor(n/4) - (1 if n even). - David Pasino, Jun 17 2016
E.g.f.: (4 - sin(x) - cos(x) + x*sinh(x) + (x - 3)*cosh(x))/4. - Ilya Gutkovskiy, Jun 21 2016
a(n) = floor((n-1)/2) - floor(n/4), n >= 0 (from the preceding a(n) formula). - Wolfdieter Lang, May 08 2017
a(n) = (2*n - 3 - 2*cos(n*Pi/2) - 3*cos(n*Pi) - 2*sin(n*Pi/2))/8. - Wesley Ivan Hurt, Oct 01 2017
a(n) = Sum_{i=1..floor((n-1)/2)} (n-i-1) mod 2. - Wesley Ivan Hurt, Nov 17 2017

More terms from Sascha Kurz, Mar 25 2002

A001972 Expansion of 1/((1-x)^2(1-x^4)) = 1/( (1+x)(1+x^2)*(1-x)^3 ).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 128, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 242, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 392, 406, 420, 435, 450, 465

Offset: 0

N. J. A. Sloane

First differences are A008621. - Amarnath Murthy, Apr 26 2004
a(n) = least k > a(n-1) such that k + a(n-1) + a(n-2) + a(n-3) is triangular. - Amarnath Murthy, Apr 26 2004
From Jon Perry, Nov 16 2010: (Start)
Column sums of the following array:
  1 2 3 4 5 6 7  8  9...
          1 2 3  4  5...
                    1...
  --------------------
  1 2 3 4 6 8 10 12 15 (End)
A001972(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2=4x+y. - Clark Kimberling, Jun 04 2012
Number of partitions of n into parts 1 (of two sorts) and 4 (of one sort). - Joerg Arndt, Aug 08 2013
In the polynomial sequence s(n) = (x*s(n-1)*s(n-4) + y*s(n-2)*s(n-3))/s(n-5), with s(k) = 1 for k = 0..4, the leading term of s(n+5) is x^a(n). See A333260. - Michael Somos, Mar 13 2020

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.
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
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]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 208
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=4]
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
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Bisections are A000217 and A007590. - Amarnath Murthy, Apr 26 2004

Cf. A001972, A002620, A008621, A056594, A130519, A333260.

[Floor((n+3)^2/8): n in [0..60]]; // Vincenzo Librandi, Aug 15 2011

A001972:=-(2-z+z**3-2*z**4+z**5)/(z+1)/(z**2+1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 1

CoefficientList[Series[1/((1-x)^2(1-x^4)),{x,0,80}],x]  (* Harvey P. Dale, Mar 27 2011 *)

PARI
```
a(n)=(n+3)^2\8;
```

From Michael Somos, Apr 21 2000: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) + 1.
a(n) = floor((n+3)^2/8). (End)
a(n) = Sum_{k=0..n} floor((k+4)/4) = n + 1 + Sum_{k=0..n} floor(k/4). - Paul Barry, Aug 19 2003
a(n) = a(n-4) + n + 1. - Paul Barry, Jul 14 2004
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+4} floor(j/4);
a(n-4) = (1/2)*floor(n/4)*(2*n - 2 - 4*floor(n/4)). (End)
A002620(n+1) = a(2*n-1)/2.
A000217(n+1) = a(2*n).
a(n)+a(n+1)+a(n+2)+a(n+3) = (n+4)*(n+5)/2. - Amarnath Murthy, Apr 26 2004
a(n) = n^2/8 + 3*n/4 + 15/16 + (-1)^n/16 + A056594(n+3)/4. - Amarnath Murthy, Apr 26 2004
a(n) = A130519(n+4). - Franklin T. Adams-Watters, Jul 10 2009
a(n) = floor((n+1)/(1-e^(-8/(n+1)))). - Richard R. Forberg, Aug 07 2013
a(n) = a(-6-n) for all n in Z. - Michael Somos, Mar 13 2020
E.g.f.: ((8 + 7*x + x^2)*cosh(x) + 2*sin(x) + (7 + 7*x + x^2)*sinh(x))/8. - Stefano Spezia, May 09 2023

Partially edited by R. J. Mathar, Jul 11 2009

A067059 Square array read by antidiagonals of partitions which half fill an n*k box, i.e., partitions of floor(nk/2) or ceiling(nk/2) into up to n positive integers, each no more than k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 4, 6, 8, 6, 4, 1, 1, 1, 1, 4, 8, 12, 12, 8, 4, 1, 1, 1, 1, 5, 10, 18, 20, 18, 10, 5, 1, 1, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 1, 6, 15, 33, 49, 58, 49, 33, 15, 6, 1, 1, 1, 1, 6

Offset: 0

Henry Bottomley, Feb 17 2002

The number of partitions of m into up to n positive integers each no more than k is maximized for given n and k by m=floor(nk/2) or ceiling(nk/2) (and possibly some other values).

			Rows start:
1, 1, 1, 1, 1, 1, ...;
1, 1, 1, 1, 1, 1, ...;
1, 1, 2, 2, 3, 3, ...;
1, 1, 2, 3, 5, 6, ...;
1, 1, 3, 5, 8, 12, ...; etc.
T(4,5)=12 since 10 can be partitioned into
5+5, 5+4+1, 5+3+2, 5+3+1+1, 5+2+2+1, 4+4+2, 4+3+3,
4+4+1+1, 4+3+2+1, 4+2+2+2, 3+3+3+1, and 3+3+2+2.

As this is symmetric, rows and columns each include A000012 twice, A008619, A001971, A001973, A001975, A001977, A001979 and A001981. Diagonal is A029895. T(n, n*(n-1)) is the magic series A052456.

A067059 := proc(n,k)
    local m,a1,a2 ;
    a1 := 0 ;
    m := floor(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a1 := a1+1 ;
            end if ;
        end if;
    end do:
    a2 := 0 ;
    m := ceil(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a2 := a2+1 ;
            end if ;
        end if;
    end do:
    max(a1,a2) ;
end proc:
for d from 0 to 12 do
    for k from 0 to d do
        printf("%d,",A067059(d-k,k)) ;
    end do:
end do: # R. J. Mathar, Nov 13 2016

t[n_, k_] := Length[ IntegerPartitions[ Floor[n*k/2], n, Range[k]]]; Flatten[ Table[ t[n-k , k], {n, 0, 13}, {k, 0, n}]] (* Jean-François Alcover, Jan 02 2012 *)

def A067059(n, k):
    return Partitions((n*k)//2, max_length=n, max_part=k).cardinality()
for n in (0..9): [A067059(n,k) for k in (0..9)] # Peter Luschny, May 05 2014

A061857 Triangle in which the k-th item in the n-th row (both starting from 1) is the number of ways in which we can add 2 distinct integers from 1 to n in such a way that the sum is divisible by k.

Original entry on oeis.org

0, 1, 0, 3, 1, 1, 6, 2, 2, 1, 10, 4, 4, 2, 2, 15, 6, 5, 3, 3, 2, 21, 9, 7, 5, 4, 3, 3, 28, 12, 10, 6, 6, 4, 4, 3, 36, 16, 12, 8, 8, 5, 5, 4, 4, 45, 20, 15, 10, 9, 7, 6, 5, 5, 4, 55, 25, 19, 13, 11, 9, 8, 6, 6, 5, 5, 66, 30, 22, 15, 13, 10, 10, 7, 7, 6, 6, 5, 78, 36, 26, 18, 16, 12, 12, 9, 8, 7, 7

Offset: 1

Antti Karttunen, May 11 2001

Since the sum of two distinct integers from 1 to n can be as much as 2n-1, this triangular table cannot show all the possible cases. For larger triangles showing all solutions, see A220691 and A220693. - Antti Karttunen, Feb 18 2013 [based on Robert Israel's mail, May 07 2012]

			The second term on the sixth row is 6 because we have 6 solutions: {1+3, 1+5, 2+4, 2+6, 3+5, 4+6} and the third term on the same row is 5 because we have solutions {1+2,1+5,2+4,3+6,4+5}.
Triangle begins:
   0;
   1,  0;
   3,  1,  1;
   6,  2,  2,  1;
  10,  4,  4,  2,  2;
  15,  6,  5,  3,  3,  2;
  21,  9,  7,  5,  4,  3,  3;
  28, 12, 10,  6,  6,  4,  4,  3;
  36, 16, 12,  8,  8,  5,  5,  4,  4;
  45, 20, 15, 10,  9,  7,  6,  5,  5,  4;

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Stackexchange, Question 142323
Index entries for sequences related to subset sums modulo m

This is the lower triangular region of square array A220691. See A220693 for all nonzero solutions.

The left edge (first diagonal) of the triangle: A000217, the second diagonal is given by C(((n+(n mod 2))/2), 2)+C(((n-(n mod 2))/2), 2) = A002620, the third diagonal by A058212, the fourth by A001971, the central column by A042963? trinv is given at A054425. Cf. A061865.

a061857 n k = length [()| i <- [2..n], j <- [1..i-1], mod (i + j) k == 0]
a061857_row n = map (a061857 n) [1..n]
a061857_tabl = map a061857_row [1..]
-- Reinhard Zumkeller, May 08 2012

[seq(DivSumChoose2Triangle(j),j=1..120)]; DivSumChoose2Triangle := (n) -> nops(DivSumChoose2(trinv(n-1),(n-((trinv(n-1)*(trinv(n-1)-1))/2))));
DivSumChoose2 := proc(n,k) local a,i,j; a := []; for i from 1 to (n-1) do for j from (i+1) to n do if(0 = ((i+j) mod k)) then a := [op(a),[i,j]]; fi; od; od; RETURN(a); end;

a[n_, 1] := n*(n-1)/2; a[n_, k_] := Module[{r}, r = Reduce[1 <= i < j <= n && Mod[i + j, k] == 0, {i, j}, Integers]; Which[Head[r] === Or, Length[r], Head[r] === And, 1, r === False, 0, True, Print[r, " not parsed"]]]; Table[a[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 04 2014 *)

(define (A061857 n) (A220691bi (A002024 n) (A002260 n))) ;; Antti Karttunen, Feb 18 2013. Needs A220691bi from A220691.

From Robert Israel, May 08 2012: (Start)
Let n+1 = b mod k with 0 <= b < k, q = (n+1-b)/k.  Let k = c mod 2, c = 0 or 1.
If b = 0 or 1 then a(n,k) = q^2*k/2 + q*b - 2*q - b + 1 + c*q/2.
If b >= (k+3)/2 then a(n,k) = q^2*k/2 + q*b - 2*q + b - 1 - k/2 + c*(q+1)/2.
Otherwise a(n,k) = q^2*k/2 + q*b - 2*q + c*q/2. (End)

Offset corrected by Reinhard Zumkeller, May 08 2012

A001973 Expansion of (1+x^3)/((1-x)(1-x^2)^2(1-x^3)).

Original entry on oeis.org

1, 1, 3, 5, 8, 12, 18, 24, 33, 43, 55, 69, 86, 104, 126, 150, 177, 207, 241, 277, 318, 362, 410, 462, 519, 579, 645, 715, 790, 870, 956, 1046, 1143, 1245, 1353, 1467, 1588, 1714, 1848, 1988

Offset: 0

N. J. A. Sloane

a(1..3)=0; a(n) is the number of partitions of 2*(n+1) with 4 different numbers from the set {1,...,n}; the number of partitions of 2*n + 2 - C and 2*n + 2 + C are equal; example: n=6; 2*n + 2 = 14; a(6)=3; (10,1), (11,1), (12,2), (13,2), (14,3), (15,2), (16,2), (17,1), (18,1). - Paul Weisenhorn, Jun 01 2009. [I believe this comment refers to the sequence 0, 0, 0, 1, 1, 3, 5, ... with offset 1. - N. J. A. Sloane, Mar 30 2023]

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, pages 110ff. [From Paul Weisenhorn, Jun 01 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..1000
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, 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
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1)

A001973:=(1-z+z**2)/(z+1)/(z**2+z+1)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+1), right=Set(U, card=2)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=6..45) ; # Zerinvary Lajos, Feb 07 2008

CoefficientList[Series[(1+x^3)/((1-x)*(1-x^2)^2*(1-x^3)),{x,0,40}],x] (* Vincenzo Librandi, Jun 11 2012 *)

Vec((1+x^3)/((1-x)*(1-x^2)^2*(1-x^3))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) is the coefficient of x^(2*n+2) in the g.f. Product_{s=1..4} (x^s - x^(n+1))/(1-x^s). - Paul Weisenhorn, Jun 01 2009

a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7). Vincenzo Librandi, Jun 11 2012

A001977 Number of partitions of 3n into n parts from the set {0, 1, ..., 6} (repetitions admissible).

Original entry on oeis.org

1, 1, 4, 8, 18, 32, 58, 94, 151, 227, 338, 480, 676, 920, 1242, 1636, 2137, 2739, 3486, 4370, 5444, 6698, 8196, 9926, 11963, 14293, 17002, 20076, 23612, 27594, 32134, 37212, 42955, 49341, 56512, 64444, 73294, 83036, 93844, 105690, 118765, 133037

Offset: 0

N. J. A. Sloane

In Cayley's terminology, this is the number of literal terms of degree n and weight 3*n involving the letters a, b, c, d, e, f, g, having weights 0, 1, 2, 3, 4, 5, 6 respectively, a number which is also equal to the coefficient of x^(3n)z^n in the development of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008

a(0..5)=0; a(n) is the number of partitions of 3*(n+1) with 6 different numbers from the set {1,...,n}; the number of partitions of 3*(n+1)-C and 3*(n+1)+C are equal; example: n=8; 3*n+3=27; a(8)=4; (21,1), (22,1),(23,2), (24,2), (25,3), (26,3), (27,4), (28,3), (29,3), (30,2), (31,2),(32,1), (33,1). - Paul Weisenhorn, Jun 01 2009

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, Band 2, Klett, 1975, pages 110- [From Paul Weisenhorn, Jun 01 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).

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

LinearRecurrence[{1, 2, -1, -1, -1, -1, 0, 2, 2, 0, -1, -1, -1, -1, 2, 1, -1}, {1, 1, 4, 8, 18, 32, 58, 94, 151, 227, 338, 480, 676, 920, 1242, 1636, 2137}, 100] (* Jean-François Alcover, Feb 25 2020 *)

f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)*(1-x^6*z)); n=200; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(n=0,60,print1(polcoeff(polcoeff(p,3*n),n)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008

a(n) is the coefficient of x^(3*n+3) from the g.f. Product_{s=1..6} (x^s-x^(n+1))/(1-x^s). - Paul Weisenhorn, Jun 01 2009

G.f.: (x^10+x^8+3*x^7+4*x^6+4*x^5+4*x^4+3*x^3+x^2+1) / ((x^2+x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) *(x+1)^3 *(x-1)^6). - Alois P. Heinz, Jul 26 2015

Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008

A001981 Restricted partitions.

Original entry on oeis.org

1, 1, 5, 13, 33, 73, 151, 289, 526, 910, 1514, 2430, 3788, 5744, 8512, 12346, 17575, 24591, 33885, 46029, 61731, 81805, 107233, 139143, 178870, 227930, 288100, 361384, 450096, 556834, 684572, 836618, 1016737, 1229093, 1478379, 1769773

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of 4n into up to 8 parts each no more than n; or partitions of 4n into up to n parts each no more than 8; or partitions of 5n into exactly n single-digit parts; or partitions of 4(n+2) into exactly 8 parts each no more than n+1; or partitions of 4(n+9) into exactly 8 distinct parts each no more than n+8; etc. Points lie on 252 different septics with the pattern repeating every 420 points, amounting to 4 sets of parallel septics depending on whether n mod 6 is in {0}, {1,5}, {2,4} or {3}.

Also, the dimension of the vector space of homogeneous covariants of degree n for the binary form of degree 8. - Leonid Bedratyuk, Dec 06 2006

			a(3)=13 since partitions of 12 into up to 8 parts each no more than 3 are 3+3+3+3 = 3+3+3+2+1 = 3+3+3+1+1+1 = 3+3+2+2+2 = 3+3+2+2+1+1 = 3+3+2+1+1+1+1 = 3+3+1+1+1+1+1+1 = 3+2+2+2+2+1 = 3+2+2+2+1+1+1 = 3+2+2+1+1+1+1+1 = 2+2+2+2+2+2 = 2+2+2+2+2+1+1 = 2+2+2+2+1+1+1+1; or equivalently partitions of 15 into exactly 3 single-digit numbers are 9+5+1 = 9+4+2 = 9+3+3 = 8+6+1 = 8+5+2 = 8+4+3 = 7+7+1 = 7+6+2 = 7+5+3 = 7+4+4 = 6+6+3 = 6+5+4 =5+5+5.

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.
Hilbert, D., Theory of algebraic invariants. Lectures. Cambridge University Press, (1993).
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).
Springer, T.A., Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, (1977).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Henry Bottomley, Partition and composition calculator.
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, 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.

a:= n-> subs({x=1}, convert(series((product('1-x^i', 'i'=9..8+n)/ product('1-x^k', 'k'=2..n)), x, 4*n+1), polynom)): seq (a(n), n=0..40); # Leonid Bedratyuk, Dec 06 2006

a[n_] := Length[IntegerPartitions[4*n, 8, Range[n]]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 17 2014 *)

a(n) =A067059(n, 8) =A067059(8, n) =(1/152409600) * (1812n^7 + 57078n^6 + 748314n^5 + 5278770n^4 + 21727272n^3 + 52982181n^2 + 77609245n + 66220839 + (297675n^2+2679075n+27088425)*(1, -1)pcr(n, 2) + (1254400*n+5644800)*(2, -1, -1)pcr(n, 3) + 9408000*(0, -1, 1)pcr(n, 3) + 4762800*(1, 1, -1, -1)pcr(n, 4) + 24385536*(1, -1, 0, 0, 0)pcr(n, 5) + 6220800(3, -1, 2, -2, 1, -3, 0)pcr(n, 7)) where for example (0, -1, 1)pcr(n, 3) means the value 0 if n mod 3 = 0, the value -1 if n mod 3 = 1 and the value 1 if n mod 3 = 2. - Henry Bottomley, Jul 19 2003

Edited by Henry Bottomley, Jul 19 2003

A047971 Triangle of coefficients of Gaussian polynomials [ n+3,3 ].

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

a(n) as illustrated is related to the following sequences: The row sum values are A001400. The column sums are A000292. The row lengths are the stuttering sequence A037915 (stutter values in A016777). The column lengths are the sequence A016777. The max values in each column are A001971. - Alford Arnold, Aug 16 2004
The Gaussian polynomial (or Gaussian binomial) [n,3]q is an example of a q-binomial coefficient (see the link) and may be defined for n >= 3 by [n,3]_q = ([n]_q * [n-1]_q * [n-2]_q)/([1]_q * [2]_q * [3]_q), where [n]_q := q^n - 1. The first few values are: [3,3]_q = 1; [4,3]_q = 1 + q + q^2 + q^3; [5,3]_q = 1 + q + 2q^2 + 2q^3 + 2q^4 + q^5 + q^6. - _Peter Bala, Sep 23 2007
The entry a(p,w), p >= 0, w = 0,1,...,3*p, of this irregular triangle is the number of nonnegative solutions of m_0 + m_1 + m_2 + m_3 = p and 1*m_1 + 2*m_2 + 3*m_3 = w. See the Hawkins reference given in A008967, p. 264, (4,7),(4.8), concerning Cayley's counting problem.  N(p,3,w) there equals a(p,w). The o.g.f. has been given in the formula section by Peter Bala.  See also the Cayley reference given in A008967, p. 110, 35. with m = 3, Theta = p and q = w. - Wolfdieter Lang, Dec 02 2012
The entry a(p,w) p >= 0, w = 0,1,...,3*p, of this array gives the number of partitions of w into at most p parts, each at most 3. This follows from the preceding comment with the two Diophantine equations. From Andrews, p. 33 and p. 35, a(p,w) (called there p(N,M,n) with N=p, M=3, n=w) gives also the number of partitions of w into at most 3 parts, each at most p. This is in accordance with the symmetry of the q-binomials [p+3,p] = [p+3,3]. - Wolfdieter Lang, Dec 04 2012

			The table a(p,w) = [q^w][p+3,3]_q starts:
p\w 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
0:  1
1:  1  1  1  1
2:  1  1  2  2  2  1  1
3:  1  1  2  3  3  3  3  2  1  1
4:  1  1  2  3  4  4  5  4  4  3  2  1  1
5:  1  1  2  3  4  5  6  6  6  6  5  4  3  2  1  1
6:  1  1  2  3  4  5  7  7  8  8  8  7  7  5  4  3  2  1  1
... Reformatted and extended by _Wolfdieter Lang_, Dec 04 2012
Partition example: Row p=2 is 1 1 2 2 2 1 1 because there are ten solution for (m_0, m_1, m_2, m_3) of the first equation given in a comment above, namely (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1) and (0,0,1,1) which have the w = 1*m_1 + 2*m_2 + 3*m_3 values 0, 2, 4, 6, 1, 2, 3, 3, 4 and 5, respectively. Therefore there are 1, 1, 2, 2, 2, 1, 1 solutions for w = 0, 1, 2, 3, 4, 5, 6, respectively. - _Wolfdieter Lang_, Dec 03 2012
a(4,5) = 4 because there are 4 partitions of 5 with 1, 2, 3  or 4 parts, each being <= 3, namely all partitions of 5 excluding 5, 14 and 11111. There are also 4 partitions of 5 with 1, 2, or 3 parts, each being <= 4, namely all partitions of 5 excluding 5, 1112 and 11111. - _Wolfdieter Lang_, Dec 04 2012
The table may also be arranged as follows (see the Alford Arnold comment above):
1
..1
..1..1
..1..1..1
..1..2..1..1
.....2..2..1..1
.....2..3..2..1..1
.....1..3..3..2..1..1

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 242.

Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A008967.

Cf. A001400.

nmax = 6;
se = Series[ 1/Product[1 - q^k*x, {k, 0, 3}], {x, 0, nmax}];
row[n_] := CoefficientList[ SeriesCoefficient[se, n], q];
Flatten[ Table[ row[n], {n, 0, nmax}]] (* Jean-François Alcover, Dec 19 2011 *)

O.g.f.: 1/((1-x)(1-qx)(1-q^2x)(1-q^3x)) = 1 + x(1 + q + q^2 + q^3) + x^2(1 + q + 2q^2 + 2q^3 + 2q^4 + q^5 + q^6) + .... - Peter Bala, Sep 23 2007

cp's OEIS Frontend

A056837 Duplicate of A001971.

Views

Author

Keywords

A000601 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^3)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A059169 Number of partitions of n into 3 parts which form the sides of a nondegenerate isosceles triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A001972 Expansion of 1/((1-x)^2*(1-x^4)) = 1/( (1+x)*(1+x^2)*(1-x)^3 ).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A067059 Square array read by antidiagonals of partitions which half fill an n*k box, i.e., partitions of floor(nk/2) or ceiling(nk/2) into up to n positive integers, each no more than k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Sage

A061857 Triangle in which the k-th item in the n-th row (both starting from 1) is the number of ways in which we can add 2 distinct integers from 1 to n in such a way that the sum is divisible by k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Scheme

Formula

Extensions

A000601 Expansion of 1/((1-x)^2(1-x^2)(1-x^3)).

A001972 Expansion of 1/((1-x)^2(1-x^4)) = 1/( (1+x)(1+x^2)*(1-x)^3 ).

A001973 Expansion of (1+x^3)/((1-x)(1-x^2)^2(1-x^3)).