A195825 -id:A195825 - cp's OEIS frontend

A210843 Level of the n-th plateau of the column k of the square array A195825, when k -> infinity.

1, 4, 13, 35, 86, 194, 415, 844, 1654, 3133, 5773, 10372, 18240, 31449, 53292, 88873, 146095, 236977, 379746, 601656, 943305, 1464501, 2252961, 3436182, 5198644, 7805248, 11634685, 17224795, 25336141, 37038139, 53828275, 77792869

Offset: 1

Omar E. Pol, Jun 19 2012

nonn

Also the first (k+1)/2 terms of this sequence are the levels of the (k+1)/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 2, if k is odd.
Also the first k/2 terms of this sequence are the levels of the k/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 3, if k is a positive even number.
For the visualization of the plateaus see the graph of the sequences mentioned in crossrefs section (columns k=1..10 of A195825), for example see the graph of A210964.
Also numbers that are repeated in column k of square array A195825, when k -> infinity.
Note that the definition and the comments related to the square array A195825 mentioned above are also valid for the square array A211970, since both arrays contains the same columns, if k >= 1.
Is this the EULER transform of 4, 3, 3, 3, 3, 3, 3...?

			Column 1 of A195825 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11... The column contains only one plateau: [1, 1] which has level 1 and length 2. So a(1) = 1.
Column 3 of A195825 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10... The column contains only two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2. So a(1)= 1 and a(2) = 2.
Column 6 of A195825 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3. So a(1) = 1, a(2) = 4 and a(3) = 13.

Partial sums of A000716. Column 3 of A210764.
Columns (k=1..10) of A195825: A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
Cf. A000070, A000712, A000713, A010815, A211970.

CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^3,{k,1,50}],{x,0,50}],x] (* Vaclav Kotesovec, Aug 16 2015 *)

From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ sqrt(2*n)/Pi * A000716(n).
a(n) ~ exp(sqrt(2*n)*Pi) / (8*Pi*n).
(End)

A195849 Column 5 of array A195825. Also column 1 of triangle A195839. Also 1 together with the row sums of triangle A195839.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 14, 16, 21, 27, 32, 34, 36, 38, 44, 54, 67, 77, 84, 88, 95, 107, 128, 152, 174, 188, 200, 215, 242, 281, 329, 370, 402, 428, 462, 513, 589, 674, 754, 816, 873, 940, 1041, 1176, 1333, 1477, 1600, 1710, 1845

Offset: 0

Omar E. Pol, Oct 07 2011

nonn

Note that this sequence contains three plateaus: [1, 1, 1, 1, 1, 1], [4, 4, 4, 4], [13, 13]. For more information see A210843. See also other columns of A195825. - Omar E. Pol, Jun 29 2012

Number of partitions of n into parts congruent to 0, 1 or 6 (mod 7). - Ludovic Schwob, Aug 05 2021

Ludovic Schwob, Table of n, a(n) for n = 0..10000

Cf. A000041, A001082, A006950, A036820, A057077, A118277, A195825, A195829, A195839, A195848, A195850, A195851, A195852, A196933, A210843, A210964, A211971.

A118277 := proc(n)
        7*n^2/8+7*n/8-3/16+3*(-1)^n*(1/16+n/8) ;
end proc:
A195839 := proc(n, k)
        option remember;
        local ks, a, j ;
        if A118277(k) > n then
                0 ;
        elif n <= 5 then
                return 1;
        elif k = 1 then
                a := 0 ;
                for j from 1 do
                        if A118277(j) <= n-1 then
                                a := a+procname(n-1, j) ;
                        else
                                break;
                        end if;
                end do;
                return a;
        else
                ks := A118277(k) ;
                (-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;
        end if;
end proc:
A195849 := proc(n)
        A195839(n+1,1) ;
end proc:
seq(A195849(n), n=0..60) ; # R. J. Mathar, Oct 08 2011

m = 61;
Product[1/((1 - x^(7k))(1 - x^(7k - 1))(1 - x^(7k - 6))), {k, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Apr 13 2020, after Ilya Gutkovskiy *)

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-1))*(1 - x^(7*k-6))). - Ilya Gutkovskiy, Aug 13 2017

a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*sin(Pi/7)*n). - Vaclav Kotesovec, Aug 14 2017

A210964 Column 10 of square array A195825. Also column 1 of triangle A210954. Also 1 together with the row sums of triangle A210954.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 194, 195

Offset: 0

Omar E. Pol, Jun 16 2012

nonn

Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13, 13], [35, 35, 35, 35, 35], [86, 86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..3000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - The asymptotic ratio
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000041, A006950, A036820, A057077, A195818, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A210944, A210954, A211970, A211971.

Cf. A247133.

nmax = 100; CoefficientList[Series[Product[1 / ((1 - x^(12*k)) * (1 - x^(12*k-1)) * (1 - x^(12*k-11))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

Expansion of 1 / f(-x, -x^11) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 10 2015
Partitions of n into parts of the form 12*k, 12*k+1, 12*k+11. - Michael Somos, Jan 10 2015
Euler transform of period 12 sequence [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, ...]. - Michael Somos, Jan 10 2015
G.f.: Product_{k>0} 1 / ((1 - x^(12*k)) * (1 - x^(12*k - 1)) * (1 - x^(12*k - 11))).
Convolution inverse of A247133.
a(n) ~ sqrt(2)*(1+sqrt(3)) * exp(Pi*sqrt(n/6)) / (8*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = (1/n)*Sum_{k=1..n} A284372(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
a(n) = a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - - (with the convention a(n) = 0 for negative n), where 1, 11, 14, 34, ... is the sequence of generalized 14-gonal numbers A195818. - Peter Bala, Dec 10 2020

A195850 Column 6 of array A195825. Also column 1 of triangle A195840. Also 1 together with the row sums of triangle A195840.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 36, 38, 44, 54, 67, 77, 83, 86, 89, 95, 107, 128, 152, 173, 186, 194, 202, 216, 242, 281, 328, 368, 396, 415, 434, 464, 514, 588, 672, 748, 803, 844

Offset: 0

Omar E. Pol, Oct 07 2011

Note that this sequence contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Number of partitions of n into parts congruent to 0, 1 or 7 (mod 8). - Peter Bala, Dec 10 2020

Cf. A000041, A001082, A006950, A036820, A057077, A074377, A195825, A195830, A195848, A195849, A195851, A195852, A196933, A210964, A211971.

G.f.: Product_{k>=1} 1/((1 - x^(8*k))*(1 - x^(8*k-1))*(1 - x^(8*k-7))). - Ilya Gutkovskiy, Aug 13 2017
a(n) ~ exp(Pi*sqrt(n)/2) / (4*sqrt(2-sqrt(2))*n). - Vaclav Kotesovec, Aug 14 2017
a(n) = a(n-1) + a(n-7) - a(n-10) - a(n-22) + + - - (with the convention a(n) = 0 for negative n), where 1, 7, 10, 22, ... is the sequence of generalized 10-gonal numbers A074377. - Peter Bala, Dec 10 2020

A195851 Column 7 of array A195825. Also column 1 of triangle A195841. Also 1 together with the row sums of triangle A195841.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 87, 89, 95, 107, 128, 152, 173, 185, 192, 196, 203, 216, 242, 281, 328, 367, 394, 409, 421, 436, 465

Offset: 0

Omar E. Pol, Oct 07 2011

nonn

Note that this sequence contains four plateaus: [1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4], [13, 13, 13, 13], [35, 35]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Cf. A000041, A001082, A006950, A036820, A057077, A195160, A195831, A195825, A195848, A195849, A195850, A195852, A196933, A210964, A211971.

A195160 := proc(n)
        (18*n*(n+1)+5*(2*n+1)*(-1)^n-5)/16 ;
end proc:
A195841 := proc(n, k)
        option remember;
        local ks, a, j ;
        if A195160(k) > n then
                0 ;
        elif n <= 5 then
                return 1;
        elif k = 1 then
                a := 0 ;
                for j from 1 do
                        if A195160(j) <= n-1 then
                                a := a+procname(n-1, j) ;
                        else
                                break;
                        end if;
                end do;
                return a;
        else
                ks := A195160(k) ;
                (-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;
        end if;
end proc:
A195851 := proc(n)
        A195841(n+1,1) ;
end proc:
seq(A195851(n), n=0..60) ; # R. J. Mathar, Oct 08 2011

G.f.: Product_{k>=1} 1/((1 - x^(9*k))*(1 - x^(9*k-1))*(1 - x^(9*k-8))). - Ilya Gutkovskiy, Aug 13 2017

a(n) ~ exp(Pi*sqrt(2*n)/3) / (8*sin(Pi/9)*n). - Vaclav Kotesovec, Aug 14 2017

A196933 Column 9 of array A195825. Also column 1 of triangle A195843. Also 1 together with the row sums of triangle A195843.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 195, 197, 203, 216, 242, 281

Offset: 0

Omar E. Pol, Oct 07 2011

nonn

Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13], [35, 35, 35, 35], [86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000041, A001082, A006950, A036820, A057077, A195313, A195825, A195833, A195848, A195849, A195850, A195851, A195852, A210964, A211971.

T := Product[1/((1 - x^(11*k))*(1 - x^(11*k - 1))*(1 - x^(11*k - 10))), {k, 1, 70}]; a:= CoefficientList[Series[T, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 28 2018 *)

G.f.: Product_{k>=1} 1/((1 - x^(11*k))*(1 - x^(11*k-1))*(1 - x^(11*k-10))). - Ilya Gutkovskiy, Aug 13 2017

a(n) ~ exp(Pi*sqrt(2*n/11)) / (8*sin(Pi/11)*n). - Vaclav Kotesovec, Aug 14 2017

More terms from Omar E. Pol, Jun 10 2012

A195852 Column 8 of array A195825. Also column 1 of triangle A195842. Also 1 together with the row sums of triangle A195842.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 194, 197, 203, 216, 242, 281, 328, 367, 393, 407

Offset: 0

Omar E. Pol, Oct 07 2011

Note that this sequence contains four plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13], [35, 35, 35]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Number of partitions of n into parts congruent to 0, 1 or 9 (mod 10). - Peter Bala, Dec 10 2020

Cf. A000041, A001082, A006950, A036820, A057077, A195162, A195825, A195832, A195848, A195849, A195850, A195851, A196933, A210964, A211971.

G.f.: Product_{k>=1} 1/((1 - x^(10*k))*(1 - x^(10*k-1))*(1 - x^(10*k-9))). - Ilya Gutkovskiy, Aug 13 2017
a(n) ~ exp(Pi*sqrt(n/5))/(2*(sqrt(5)-1)*n). - Vaclav Kotesovec, Aug 14 2017
a(n) = a(n-1) + a(n-9) - a(n-12) - a(n-28) + + - - (with the convention a(n) = 0 for negative n), where 1, 9, 12, 28, ... is the sequence of generalized 12-gonal numbers A195162. - Peter Bala, Dec 10 2020

More terms from Omar E. Pol, Jun 10 2012

A006950 G.f.: Product_{k>=1} (1 + x^(2k - 1)) / (1 - x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 196, 236, 287, 350, 420, 501, 602, 722, 858, 1016, 1206, 1431, 1687, 1981, 2331, 2741, 3206, 3740, 4368, 5096, 5922, 6868, 7967, 9233, 10670, 12306, 14193, 16357, 18803, 21581

Offset: 0

N. J. A. Sloane, Warren D. Smith

nonn

Also the number of partitions of n in which all odd parts are distinct. There is no restriction on the even parts. E.g., a(9)=13 because "9 = 8+1 = 7+2 = 6+3 = 6+2+1 = 5+4 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 4+2+2+1 = 3+2+2+2 = 2+2+2+2+1". - Noureddine Chair, Feb 03 2005
Number of partitions of n in which each even part occurs with even multiplicity. There is no restriction on the odd parts.
Also the number of partitions of n into parts not congruent to 2 mod 4. - James Sellers, Feb 08 2002
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras o(n) of skew-symmetric n X n matrices, n=0,1,2,3,... (the cases n=0,1 being degenerate). This sequence, A015128 and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=2.
Equals polcoeff inverse of A010054 with alternate signs. - Gary W. Adamson, Mar 15 2010
It appears that this sequence is related to the generalized hexagonal numbers (A000217) in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: this is 1 together with the row sums of triangle A195836, also column 1 of A195836, also column 2 of the square array A195825. - Omar E. Pol, Oct 09 2011
Since this is also column 2 of A195825 so the sequence contains only one plateau [1, 1, 1] of level 1 and length 3. For more information see A210843. - Omar E. Pol, Jun 27 2012
Convolution of A035363 and A000700. - Vaclav Kotesovec, Aug 17 2015
Also the number of ways to stack n triangles in a valley (pointing upwards or downwards depending on row parity). - Seiichi Manyama, Jul 07 2018

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...
G.f. = q^-1 + q^7 + q^15 + 2*q^23 + 3*q^31 + 4*q^39 + 5*q^47 + 7*q^55 + 10*q^63 + ...
From _Seiichi Manyama_, Jul 07 2018: (Start)
n | the ways to stack n triangles in a valley
--+------------------------------------------------------
1 | *---*
  |  \ /
  |   *
  |
2 |   *
  |  / \
  | *---*
  |  \ /
  |   *
  |
3 |   *---*     *---*
  |  / \ /       \ / \
  | *---*         *---*
  |  \ /           \ /
  |   *             *
  |
4 |     *                       *
  |    / \                     / \
  |   *---*     *---*---*     *---*
  |  / \ /       \ / \ /       \ / \
  | *---*         *---*         *---*
  |  \ /           \ /           \ /
  |   *             *             *
  |
5 |     *---*         *         *         *---*
  |    / \ /         / \       / \         \ / \
  |   *---*     *---*---*     *---*---*     *---*
  |  / \ /       \ / \ /       \ / \ /       \ / \
  | *---*         *---*         *---*         *---*
  |  \ /           \ /           \ /           \ /
  |   *             *             *             *
  |
6 |       *
  |      / \
  |     *---*         *---*     *   *     *---*
  |    / \ /         / \ /     / \ / \     \ / \
  |   *---*     *---*---*     *---*---*     *---*---*
  |  / \ /       \ / \ /       \ / \ /       \ / \ /
  | *---*         *---*         *---*         *---*
  |  \ /           \ /           \ /           \ /
  |   *             *             *             *
  |   *
  |  / \
  | *---*
  |  \ / \
  |   *---*
  |    \ / \
  |     *---*
  |      \ /
  |       *
(End)

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.
M. D. Hirschhorn, The Power of q, Springer, 2017. See pod, page 297.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Luca Ferrari, Schröder partitions, Schröder tableaux and weak poset patterns, arXiv:1606.06624 [math.CO], 2016. Mentions this sequence.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See Example 4.2 p. 13.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
L. Wang, New Congruences for Partitions where the Odd Parts are Distinct, J. Int. Seq. 18 (2015) # 15.4.2.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012

See also Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cf. A015128, A046682, A106459.
Cf. A163203.
Cf. A010054, A085642, A316384.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 06 2013

CoefficientList[ Series[ Product[(1 + x^(2k - 1))/(1 - x^(2k)), {k, 25}], {x, 0, 50}], x] (* Robert G. Wilson v, Jun 28 2012 *)
CoefficientList[Series[x*QPochhammer[-1/x, x^2] / ((1+x)*QPochhammer[x^2, x^2]), {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
CoefficientList[Series[2*(-x)^(1/8) / EllipticTheta[2, 0, Sqrt[-x]], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-Mod[i, 2]]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz *)

{a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, (-1)^(m-d)*d)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Jul 22 2009
(GW-BASIC)
' A program with two A-numbers (Note that here A000217 are the generalized hexagonal numbers):
10 Dim A000217(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 51: For j = 1 to n
30 If A000217(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A000217(j))
40 Next j: Print a(n-1);: Next n ' Omar E. Pol, Jun 10 2012

a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*A002129(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Feb 05 2002
G.f.: 1/Sum_{k>=0} (-x)^(k*(k+1)/2). - Vladeta Jovovic, Sep 22 2002 [corrected by Vaclav Kotesovec, Aug 17 2015]
a(n) = A059777(n-1)+A059777(n), n > 0. - Vladeta Jovovic, Sep 22 2002
G.f.: Product_{m>=1} (1+x^m)^(if A001511(m) > 1, A001511(m)-1 else A001511(m)). - Jon Perry, Apr 15 2005
Expansion of 1 / psi(-x) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(1/8) * eta(q^2) / (eta(q) * eta(q^4)) in powers of q.
Convolution inverse of A106459. - Michael Somos, Nov 02 2005
G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d] * x^n/n ). - Paul D. Hanna, Jul 22 2009
a(n) ~ (8*n+1) * cosh(sqrt(8*n-1)*Pi/4) / (16*sqrt(2)*n^2) - sinh(sqrt(8*n-1)*Pi/4) / (2*Pi*n^(3/2)) ~ exp(Pi*sqrt(n/2))/(4*sqrt(2)*n) * (1 - (2/Pi + Pi/16)/sqrt(2*n) + (3/16 + Pi^2/1024)/n). - Vaclav Kotesovec, Aug 17 2015, extended Jan 09 2017
Can be computed recursively by Sum_{j>=0} (-1)^(ceiling(j/2)) a(n - j(j+1)/2) = 0, for n > 0. [Merca, Theorem 4.3] - Eric M. Schmidt, Sep 21 2017
a(n) = A000041(n) - A085642(n), for n >= 1. - Wouter Meeussen, Dec 20 2017

G.f. and more terms from Vladeta Jovovic, Feb 05 2002

A036820 Number of partitions satisfying (cn(2,5) = cn(3,5) = 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 4, 5, 7, 10, 12, 14, 16, 21, 27, 33, 37, 44, 54, 68, 80, 92, 106, 129, 155, 182, 207, 240, 283, 337, 389, 444, 508, 594, 692, 797, 902, 1030, 1187, 1373, 1564, 1770, 2004, 2295, 2624, 2978, 3349, 3783, 4293, 4880, 5501, 6174, 6932, 7830, 8834

Offset: 0

Olivier Gérard

nonn

For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: (2=3 := 0).
It appears that this sequence is related to the generalized heptagonal numbers A085787 in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: Column 1 of triangle A195837. Also 1 together with the row sums of triangle A195837. Also column 3 of the square array A195825. - Omar E. Pol, Oct 08 2011
Note that this sequence contains two plateaus: [1, 1, 1, 1] and [4, 4]. For more information see A195825 and A210843. - Omar E. Pol, Jun 23 2012

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 10*x^10 + ...
G.f. = q^-9 + q^31 + q^71 + q^111 + 2*q^151 + 3*q^191 + 4*q^231 + 4*q^271 + 5*q^311 + ... - _Michael Somos_, Sep 08 2012

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A113429.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[1, 1, 0, 0, 1]
      [1+irem(d, 5)], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 04 2014

a[n_] := a[n] = If[n == 0, 1, Sum[ Sum[ d*{1, 1, 0, 0, 1}[[1 + Mod[d, 5]]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 15 2015, after Alois P. Heinz *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, (n+4)\5, (1 - x^(5*k - 4)) * (1 - x^(5*k - 1)) * (1 - x^(5*k)), 1 + x * O(x^n)), n))}; /* Michael Somos, Feb 09 2012 */
(GW-BASIC)' A program with two A-numbers:
10 Dim A085787(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 56: For j = 1 to n
30 If A085787(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A085787(j))
40 Next j: Print a(n-1);: Next n ' Omar E. Pol, Jun 10 2012

Euler transform of period 5 sequence [1, 0, 0, 1, 1, ...]. - Michael Somos, Feb 09 2012
Expansion of 1 / f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 08 2012
Convolution inverse of A113429. - Michael Somos, Feb 09 2012
G.f.: 1 / (Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 1)) * (1 - x^(5*k - 4))). - Michael Somos, Sep 08 2012
G.f.: 1 / (Sum_{k in Z} (-1)^k * x^(k * (5*k + 3) / 2)). - Michael Somos, Sep 08 2012
a(n) ~ sqrt(1+sqrt(5)) * exp(sqrt(2*n/5)*Pi) / (2^(5/2)*5^(1/4)*n). - Vaclav Kotesovec, Oct 06 2015
a(n) = (1/n)*Sum_{k=1..n} A284361(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A195848 Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 5, 7, 10, 12, 13, 14, 16, 21, 27, 32, 35, 38, 44, 54, 67, 78, 86, 94, 107, 128, 153, 176, 194, 213, 241, 282, 331, 376, 415, 456, 512, 590, 680, 767, 845, 928, 1037, 1180, 1345, 1506, 1657, 1818, 2020, 2278, 2570, 2862, 3142, 3442

Offset: 0

Omar E. Pol, Sep 24 2011

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also column 4 of A195825, therefore this sequence contains two plateaus: [1, 1, 1, 1, 1], [4, 4, 4]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 26 2012
The number of partitions of n into parts congruent to 0, 1 or 5 ( mod 6 ). - Peter Bala, Dec 09 2020

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ...
G.f. = 1/q + q^2 + q^5 + q^8 + q^11 + 2*q^14 + 3*q^17 + 4*q^20 + 4*q^23 + 4*q^26 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Bringmann, J. Lovejoy, and K. Mahlburg, A partition identity and the universal mock theta function g_2(x;q), Mathematical Research Letters, 23 (2016), 67-80.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Column 1 of triangle A195838. Also 1 together with the row sums of triangle A195838. Column 4 of array A195825.
Cf. A000041, A001082, A006950, A036820, A057077, A195825, A195828, A195849, A195850, A195851, A195852, A196933, A210843, A210964, A211971.
Cf. A089802.

A001082 := proc(n)
        if type(n,'even') then
                n*(3*n-4)/4 ;
        else
                (n-1)*(3*n+1)/4 ;
        end if;
end proc:
A195838 := proc(n,k)
        option remember;
        local ks,a,j ;
        if A001082(k+1) > n then
                0 ;
        elif n <= 5 then
                return 1;
        elif k = 1 then
                a := 0 ;
                for j from 1 do
                        if A001082(j+1) <= n-1 then
                                a := a+procname(n-1,j) ;
                        else
                                break;
                        end if;
                end do;
                return a;
        else
                ks := A001082(k+1) ;
                (-1)^floor((k-1)/2)*procname(n-ks+1,1) ;
        end if;
end proc:
A195848 := proc(n)
        A195838(n+1,1) ;
end proc:
seq(A195848(n),n=0..60) ; # R. J. Mathar, Oct 07 2011

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]^2), {x, 0, n}]; (* Michael Somos, Oct 18 2014 *)
a[ n_] := SeriesCoefficient[ 2 q^(3/8) / (QPochhammer[ q, q^2] EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* Michael Somos, Oct 18 2014 *)
nmax = 60; CoefficientList[Series[Product[(1+x^k) / ((1+x^(3*k)) * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}; /* Michael Somos, Jun 07 2012 */
From Omar E. Pol, Jun 10 2012: (Start)
(GW-BASIC)' A program with two A-numbers:
10 Dim A001082(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 58: For j = 1 to n
30 If A001082(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A001082(j))
40 Next j: Print a(n-1);: Next n (End)

Expansion of 1 / (psi(x^3) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Jun 07 2012
Expansion of q^(1/3) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)^2) in powers of q. - Michael Somos, Jun 07 2012
Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, 1, ...]. - Michael Somos, Oct 18 2014
Convolution inverse of A089802. - Michael Somos, Oct 18 2014
a(n) ~ exp(Pi*sqrt(n/3))/(4*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = (1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
From Peter Bala, Dec 09 2020: (Start)
O.g.f.: 1/( Product_{n >= 1} (1 - x^(6*n-5))*(1 - x^(6*n-1))*(1 - x^(6*n)) ).
a(n) = a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - - ... (with the convention a(n) = 0 for negative n), where 1, 5, 8, 16, ... is the sequence of generalized octagonal numbers A001082. (End)

New sequence name from Michael Somos, Oct 18 2014

cp's OEIS Frontend

A210843 Level of the n-th plateau of the column k of the square array A195825, when k -> infinity.

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A195849 Column 5 of array A195825. Also column 1 of triangle A195839. Also 1 together with the row sums of triangle A195839.

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A210964 Column 10 of square array A195825. Also column 1 of triangle A210954. Also 1 together with the row sums of triangle A210954.

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A195850 Column 6 of array A195825. Also column 1 of triangle A195840. Also 1 together with the row sums of triangle A195840.

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A195851 Column 7 of array A195825. Also column 1 of triangle A195841. Also 1 together with the row sums of triangle A195841.

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A196933 Column 9 of array A195825. Also column 1 of triangle A195843. Also 1 together with the row sums of triangle A195843.

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A195852 Column 8 of array A195825. Also column 1 of triangle A195842. Also 1 together with the row sums of triangle A195842.

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A006950 G.f.: Product_{k>=1} (1 + x^(2*k - 1)) / (1 - x^(2*k)).

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A006950 G.f.: Product_{k>=1} (1 + x^(2k - 1)) / (1 - x^(2k)).