A113429 -id:A113429 - cp's OEIS frontend

A085787 Generalized heptagonal numbers: m(5m - 3)/2, m = 0, +-1, +-2 +-3, ...

0, 1, 4, 7, 13, 18, 27, 34, 46, 55, 70, 81, 99, 112, 133, 148, 172, 189, 216, 235, 265, 286, 319, 342, 378, 403, 442, 469, 511, 540, 585, 616, 664, 697, 748, 783, 837, 874, 931, 970, 1030, 1071, 1134, 1177, 1243, 1288, 1357, 1404, 1476, 1525, 1600, 1651, 1729

Offset: 0

Jon Perry, Jul 23 2003

Zero together with the partial sums of A080512. - Omar E. Pol, Sep 10 2011
Second heptagonal numbers (A147875) and positive terms of A000566 interleaved. - Omar E. Pol, Aug 04 2012
These numbers appear in a theta function identity. See the Hardy-Wright reference, Theorem 355 on p. 284. See the g.f. of A113429. - Wolfdieter Lang, Oct 28 2016
Characteristic function is A133100. - Michael Somos, Jan 30 2017
40*a(n) + 9 is a square. - Bruno Berselli, Apr 18 2018
Numbers k such that the concatenation k225 is a square. - Bruno Berselli, Nov 07 2018
The sequence terms occur as exponents in the expansion of Sum_{n >= 0} q^(n*(n+1)) * Product_{k >= n+1} 1 - q^k = 1 - q - q^4 + q^7 + q^13 - q^18 - q^27 + + - -  ... (see Hardy and Wright, Theorem 363, p. 290). - Peter Bala, Dec 15 2024

			From the first formula: a(5) = A000217(5) + A000217(2) = 15 + 3 = 18.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Column 3 of A195152.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), this sequence (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Cf. A001622, A080512, A113429, A133100.

a085787 n = a085787_list !! n
a085787_list = scanl (+) 0 a080512_list
-- Reinhard Zumkeller, Apr 06 2015

[5*n*(n+1)/8-1/16+(-1)^n*(2*n+1)/16: n in [0..60]]; // Vincenzo Librandi, Sep 11 2011

Select[Table[(n*(n+1)/2-1)/5,{n,500}],IntegerQ] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *)

t(n)=n*(n+1)/2
for(i=0,40,print1(t(i)+t(floor(i/2)), ", "))

{a(n) = (5*(-n\2)^2 - (-n\2)*3*(-1)^n) / 2}; /* Michael Somos, Oct 17 2006 */

a(n) = A000217(n) + A000217(floor(n/2)).
a(2*n-1) = A000566(n).
a(2*n) = A147875(n). - Bruno Berselli, Apr 18 2018
G.f.: x * (1 + 3*x + x^2) / ((1 - x) * (1 - x^2)^2). a(n) = a(-1-n) for all n in Z. - Michael Somos, Oct 17 2006
a(n) = 5*n*(n + 1)/8 - 1/16 + (-1)^n*(2*n + 1)/16. - R. J. Mathar, Jun 29 2009
a(n) = (A000217(n) + A001082(n))/2 = (A001318(n) + A118277(n))/2. - Omar E. Pol, Jan 11 2013
a(n) = A002378(n) - A001318(n). - Omar E. Pol, Oct 23 2013
Sum_{n>=1} 1/a(n) = 10/9 + (2*sqrt(1 - 2/sqrt(5))*Pi)/3. - Vaclav Kotesovec, Oct 05 2016
E.g.f.: (x*(9 + 5*x)*exp(x) - (1 - 2*x)*sinh(x))/8. - Franck Maminirina Ramaharo, Nov 07 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5)/3 - 10/9 - 2*sqrt(5)*log(phi)/3, where phi is the golden ratio (A001622). - Amiram Eldar, Feb 28 2022

New name from T. D. Noe, Apr 21 2006

Formula in sequence name added by Omar E. Pol, May 28 2012

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

A133100 Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Sep 11 2007

nonn

			G.f. = 1 + x + x^4 + x^7 + x^13 + x^18 + x^27 + x^34 + x^46 + x^55 + x^70 + ...
G.f. = q^9 + q^49 + q^169 + q^289 + q^529 + q^729 + q^1089 + q^1369 + q^1849 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for characteristic functions.

Cf. f(x, x^k) for k=2..11: A080995, A010054, A133100, A089801, A274179, A214263, A281814, A205988, A281815, A186742.

Cf. A085787, A113429.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SquaresR[ 1, 40 n + 9] / 2; (* Michael Somos, Jan 30 2017 *)
a[ n_] := If[n < 0, 0, Boole @ IntegerQ @ Sqrt @ (40 n + 9)]; (* Michael Somos, Jan 30 2017 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1,n, 1 + x^k*[-1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};

PARI
```
{a(n) = issquare( 40*n + 9)};
```

f(x,x^m) = 1 + Sum_{k=1..oo} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017
The characteristic function of A085787 generalized heptagonal numbers.
Euler transform of period 10 sequence [1, -1, 0, 1, -1, 1, 0, -1, 1, -1, ...].
G.f.: Prod_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)) = Sum_{k in Z} x^((5*k^2 + 3*k) / 2).
a(n) = |A113429(n)|. a(3*n + 2) = 0.
Sum_{k=1..n} a(k) ~ 2 * sqrt(2/5) * sqrt(n). - Amiram Eldar, Jan 13 2024

A113428 Expansion of f(-x^2, -x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 0, -1, -1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Oct 31 2005

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

See the Hardy-Wright reference for the identity given as g.f. formula below. - Wolfdieter Lang, Oct 28 2016

			G.f. = 1 - x^2 - x^3 + x^9 + x^11 - x^21 - x^24 + x^38 + x^42 - x^60 - x^65 + ...
G.f. = q - q^81 - q^121 + q^361 + q^441 - q^841 - q^961 + q^1521 + q^1681 + ...

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 356, p. 284.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Cf. A003114, A057569, A090771, A010815, A113429.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Jan 06 2016 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k * [1, 0, 1, 1, 0][k%5 + 1], 1 + x * O(x^n)), n))};

Expansion of G(x) * f(-x) in powers of x where G() is the g.f. of A003114.
Euler transform of period 5 sequence [ 0, -1, -1, 0, -1, ...].
|a(n)| is the characteristic function of the numbers in A057569.
The exponents in the q-series q * A(q^40) are the square of the numbers in A090771.
G.f.: Sum_{k in Z} (-1)^k * x^((5*k^2 + k)/2) = Prod_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)).
Convolution of A003114 and A010815.
From Wolfdieter Lang, Oct 30 2016: (Start)
a(n) = (-1)^k if n = b(2*k+1) for k >= 0, a(n) = (-1)^k if n = b(2*k), for k >= 1, and a(n) = 0 otherwise, where b(n) = A057569(n). See the third formula.
G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+1)/2)*(1-x^(2*(2*n+1))). See the Hardy reference, p. 93, eq. (6.11.1) with k=2, a=x and C_n = 1.
(End)
G.f.: Sum_{n >= 0} x^(n^2)*Product_{k >= n+1} 1 - x^k. Cf. A113429. - Peter Bala, Feb 12 2021

A284361 a(n) = Sum_{d|n, d = 0, 1, or 4 mod 5} d.

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 1, 5, 10, 16, 12, 11, 1, 15, 21, 21, 1, 16, 20, 40, 22, 12, 1, 35, 31, 27, 10, 19, 30, 67, 32, 21, 12, 35, 41, 56, 1, 20, 40, 80, 42, 42, 1, 60, 75, 47, 1, 51, 50, 91, 52, 31, 1, 70, 72, 75, 20, 30, 60, 151, 62, 32, 31, 85, 71, 84, 1, 39, 70, 135

Offset: 1

Seiichi Manyama, Mar 25 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A036820 (1/f(-x, -x^4)), A113429 (f(-x, -x^4)), A102753.

Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), this sequence (k=5), A284362 (k=6), A284363 (k=7), A284372 (k=12).

Table[Sum[If[Mod[d, 5]<2 || Mod[d, 5]==4, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
Table[Total[Select[Divisors[n],MemberQ[{0,1,4},Mod[#,5]]&]],{n,70}] (* Harvey P. Dale, Aug 02 2020 *)

a(n) = sumdiv(n, d, ((d + 1) % 5 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/20 = A102753 / 10 = 0.4934802... . - Amiram Eldar, Apr 12 2024

A259358 Expansion of f(-x^5)^2 / f(-x^2, -x^3) in powers of x where f(,) is the Ramanujan general theta function.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 2, 2, 3, 2, 4, 4, 5, 4, 6, 5, 7, 7, 8, 8, 11, 10, 12, 13, 15, 14, 18, 17, 21, 21, 24, 25, 29, 29, 34, 35, 39, 40, 47, 47, 53, 55, 61, 63, 72, 73, 82, 86, 94, 97, 109, 112, 124, 129, 141, 147, 162, 167, 183, 192, 208, 217, 237

Offset: 0

Michael Somos, Jun 24 2015

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

			G.f. = 1 + x^2 + x^3 + x^4 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + ...
G.f. = q^47 + q^287 + q^407 + q^527 + 2*q^767 + q^887 + 2*q^1007 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, 2012, see p. 12, Entry 2.1.3, Equation (2.1.22).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 4.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035959, A113429, A259357.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, -1, -1, 0][k%5+1]), n))};

Expansion of f(-x^5) * f(-x, -x^4) / f(-x) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of f(-x^5) * H(x) in powers of x where f() is a Ramanujan theta funcation and H() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015
Euler transform of period 5 sequence [ 0, 1, 1, 0, -1, ...].
G.f.: Product_{k>0} (1 - x^(5*k)) / ((1 - x^(5*k-3)) * (1 - x^(5*k-2))).
Convolution of A035959 and A113429.

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^jx^(kj*(j-1)/2 + j^2).

Original entry on oeis.org

1, 1, -2, 1, -1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 2, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -2, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0

Offset: 0

Ilya Gutkovskiy, Aug 13 2017

			Square array begins:
   1,   1,   1,   1,   1,   1, ...
  -2,  -1,  -1,  -1,  -1,  -1, ...
   0,  -1,   0,   0,   0,   0, ...
   0,   0,  -1,   0,   0,   0, ...
   2,   0,   0,  -1,   0,   0, ...
   0,   1,   0,   0,  -1,   0, ...

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Columns k=0..10 give A002448, A010815, A106459, A113429, A089802, A232714, A244525, A281814 (absolute values), A205988 (absolute values), A281815 (absolute values), A247133.

Cf. A000041, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A006950, A015128, A036820, A051682, A051624, A051865, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A211970.

Table[Function[k, SeriesCoefficient[Sum[(-1)^i x^(k i (i - 1)/2 + i^2), {i, -n, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[(1 - x^((k + 2) i)) (1 - x^((k + 2) i - 1)) (1 - x^((k + 2) i - k - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(2 + k) QPochhammer[1/x, x^(2 + k)] QPochhammer[x^(-1 - k), x^(2 + k)] QPochhammer[x^(2 + k), x^(2 + k)])/((-1 + x) (-1 + x^(1 + k))), {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column 0: Sum_{j=-inf..inf} (-1)^j*x^A000290(j) = Product_{i>=1} (1 + x^i)/(1 - x^i) (convolution inverse of A015128).
G.f. of column 1: Sum_{j=-inf..inf} (-1)^j*x^A000326(j) = Product_{i>=1} (1 - x^i) (convolution inverse of A000041).
G.f. of column 2: Sum_{j=-inf..inf} (-1)^j*x^A000384(j) = Product_{i>=1} (1 - x^(2*i))/(1 + x^(2*i-1)) (convolution inverse of A006950).
G.f. of column 3: Sum_{j=-inf..inf} (-1)^j*x^A000566(j) = Product_{i>=1} (1 - x^(5*i))*(1 - x^(5*i-1))*(1 - x^(5*i-4)) (convolution inverse of A036820).
G.f. of column 4: Sum_{j=-inf..inf} (-1)^j*x^A000567(j) = Product_{i>=1} (1 - x^(6*i))*(1 - x^(6*i-1))*(1 - x^(6*i-5)) (convolution inverse of A195848).
G.f. of column 5: Sum_{j=-inf..inf} (-1)^j*x^A001106(j) = Product_{i>=1} (1 - x^(7*i))*(1 - x^(7*i-1))*(1 - x^(7*i-6)) (convolution inverse of A195849).
G.f. of column 6: Sum_{j=-inf..inf} (-1)^j*x^A001107(j) = Product_{i>=1} (1 - x^(8*i))*(1 - x^(8*i-1))*(1 - x^(8*i-7)) (convolution inverse of A195850).
G.f. of column 7: Sum_{j=-inf..inf} (-1)^j*x^A051682(j) = Product_{i>=1} (1 - x^(9*i))*(1 - x^(9*i-1))*(1 - x^(9*i-8)) (convolution inverse of A195851).
G.f. of column 8: Sum_{j=-inf..inf} (-1)^j*x^A051624(j) = Product_{i>=1} (1 - x^(10*i))*(1 - x^(10*i-1))*(1 - x^(10*i-9)) (convolution inverse of A195852).
G.f. of column 9: Sum_{j=-inf..inf} (-1)^j*x^A051865(j) = Product_{i>=1} (1 - x^(11*i))*(1 - x^(11*i-1))*(1 - x^(11*i-10)) (convolution inverse of A196933).
G.f. of column k: Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2+j^2) = Product_{i>=1} (1 - x^((k+2)*i))*(1 - x^((k+2)*i-1))*(1 - x^((k+2)*i-k-1)).

cp's OEIS Frontend

A085787 Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ...

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Extensions

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

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Maple

Mathematica

PARI

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A133100 Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

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Mathematica

PARI

PARI

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A113428 Expansion of f(-x^2, -x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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Mathematica

PARI

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A284361 a(n) = Sum_{d|n, d = 0, 1, or 4 mod 5} d.

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Mathematica

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A259358 Expansion of f(-x^5)^2 / f(-x^2, -x^3) in powers of x where f(,) is the Ramanujan general theta function.

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PARI

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A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2 + j^2).

A085787 Generalized heptagonal numbers: m(5m - 3)/2, m = 0, +-1, +-2 +-3, ...

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^jx^(kj*(j-1)/2 + j^2).