A195852 -id:A195852 - cp's OEIS frontend

A195825 Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 7, 3, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 15, 5, 3, 1, 1, 1, 1, 1, 22, 7, 4, 2, 1, 1, 1, 1, 1, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1, 56, 16, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 77, 21, 10, 4

Offset: 0

Omar E. Pol, Sep 24 2011

In the infinite square array the column k is related to the generalized m-gonal numbers, where m = k+4. For example: the first column is related to the generalized pentagonal numbers A001318. The second column is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on ... (see the program in which A195152 is a table of generalized m-gonal numbers).
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041 (see below the first row of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A195825
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.                                          with a(0)=1]
========================================================
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
Conjecture: if k is odd then column k contains (k+1)/2 plateaus whose levels are the first (k+1)/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 2. Otherwise, if k is even then column k contains k/2 plateaus whose levels are the first k/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 3. The sequence A210843 gives the levels of the plateaus of column k, when k -> infinity. For the visualization of the plateaus see the graph of a column, for example see the graph of A210964. - Omar E. Pol, Jun 21 2012

			Array begins:
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    2,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    3,  2,  1,  1,  1,  1,  1,  1,  1,  1, ...
    5,  3,  2,  1,  1,  1,  1,  1,  1,  1, ...
    7,  4,  3,  2,  1,  1,  1,  1,  1,  1, ...
   11,  5,  4,  3,  2,  1,  1,  1,  1,  1, ...
   15,  7,  4,  4,  3,  2,  1,  1,  1,  1, ...
   22, 10,  5,  4,  4,  3,  2,  1,  1,  1, ...
   30, 13,  7,  4,  4,  4,  3,  2,  1,  1, ...
   42, 16, 10,  5,  4,  4,  4,  3,  2,  1, ...
   56, 21, 12,  7,  4,  4,  4,  4,  3,  2, ...
   77, 28, 14, 10,  5,  4,  4,  4,  4,  3, ...
  101, 35, 16, 12,  7,  4,  4,  4,  4,  4, ...
  135, 43, 21, 13, 10,  5,  4,  4,  4,  4, ...
  176, 55, 27, 14, 12,  7,  4,  4,  4,  4, ...
  ...
Column 1 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.
Column 3 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10, ... The column contains two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2.
Column 6 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.

Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Columns (1-10): A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A211970.
Cf. A057077, A195152, A210843.

Column k is asymptotic to exp(Pi*sqrt(2*n/(k+2))) / (8*sin(Pi/(k+2))*n). - Vaclav Kotesovec, Aug 14 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

A211971 Column 0 of square array A211970 (in which column 1 is A000041).

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 16, 24, 36, 54, 78, 112, 160, 224, 312, 432, 590, 802, 1084, 1452, 1936, 2568, 3384, 4440, 5800, 7538, 9758, 12584, 16160, 20680, 26376, 33520, 42468, 53644, 67552, 84832, 106246, 132706, 165344, 205512, 254824, 315256, 389168, 479368

Offset: 0

Omar E. Pol, Jun 10 2012

nonn

Partial sums give A015128. - Omar E. Pol, Jan 09 2014

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A006950, A008794, A036820, A057077, A195152, A195848, A195849, A195850, A195851, A195852, A196933, A195825, A210964, A277643.

Flatten[{1, Differences[Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 60}]]}] (* Vaclav Kotesovec, Oct 25 2016 *)
CoefficientList[Series[(1 - x)/EllipticTheta[4, 0, x], {x, 0, 43}], x] (* Robert G. Wilson v, Mar 06 2018 *)

a(n) ~ exp(Pi*sqrt(n))*Pi / (16*n^(3/2)) * (1 - (3/Pi + Pi/4)/sqrt(n) + (3/2 + 3/Pi^2+ Pi^2/24)/n). - Vaclav Kotesovec, Oct 25 2016, extended Nov 04 2016

G.f.: (1 - x)/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Mar 05 2018

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

Original entry on oeis.org

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

A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 1, 1, 1, 10, 5, 2, 1, 1, 1, 16, 7, 3, 1, 1, 1, 1, 24, 11, 4, 2, 1, 1, 1, 1, 36, 15, 5, 3, 1, 1, 1, 1, 1, 54, 22, 7, 4, 2, 1, 1, 1, 1, 1, 78, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 112, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Jun 10 2012

In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A211970
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.        (if k>=1)                         with a(0)=1,
.                                          if k >= 0]
========================================================
0    4    A008794        -         -         A211971
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014

			Array begins:
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
2,     2,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
4,     3,   2,   1,   1,   1,  1,  1,  1,  1,  1, ...
6,     5,   3,   2,   1,   1,  1,  1,  1,  1,  1, ...
10,    7,   4,   3,   2,   1,  1,  1,  1,  1,  1, ...
16,   11,   5,   4,   3,   2,  1,  1,  1,  1,  1, ...
24,   15,   7,   4,   4,   3,  2,  1,  1,  1,  1, ...
36,   22,  10,   5,   4,   4,  3,  2,  1,  1,  1, ...
54,   30,  13,   7,   4,   4,  4,  3,  2,  1,  1, ...
78,   42,  16,  10,   5,   4,  4,  4,  3,  2,  1, ...
112,  56,  21,  12,   7,   4,  4,  4,  4,  3,  2, ...
160,  77,  28,  14,  10,   5,  4,  4,  4,  4,  3, ...
224, 101,  35,  16,  12,   7,  4,  4,  4,  4,  4, ...
312, 135,  43,  21,  13,  10,  5,  4,  4,  4,  4, ...
432, 176,  55,  27,  14,  12,  7,  4,  4,  4,  4, ...
...

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem

Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A195825.
Cf. A057077, A195152.

T(n,k) = A211971(n), if k = 0.

T(n,k) = A195825(n,k), if k >= 1.

cp's OEIS Frontend

A195825 Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.

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

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A211971 Column 0 of square array A211970 (in which column 1 is A000041).

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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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