A255961 -id:A255961 - cp's OEIS frontend

A255613 G.f.: Product_{k>=1} 1/(1-x^k)^(6*k).

1, 6, 33, 146, 588, 2160, 7459, 24354, 76071, 228420, 663177, 1868220, 5124224, 13718748, 35932278, 92242982, 232473006, 575971494, 1404600837, 3375138816, 7998932769, 18712911214, 43246451181, 98799885342, 223269183076, 499357990254, 1105934610042

Offset: 0

Vaclav Kotesovec, Feb 28 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
Eric Weisstein's World of Mathematics, Plane Partition
Wikipedia, Plane partition

Cf. A000219, A161870, A255610, A255611, A255612, A255614, A193427.

Column k=6 of A255961.

a:= proc(n) option remember; `if`(n=0, 1, 6*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(6*k),{k,1,nmax}],{x,0,nmax}],x]

G.f.: Product_{k>=1} 1/(1-x^k)^(6*k).
a(n) ~ 2^(1/6) * Zeta(3)^(1/3) * exp(1/2 + 2^(-1/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^6 * 3^(1/6) * sqrt(Pi) * n^(5/6)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015
G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

New name from Vaclav Kotesovec, Mar 12 2015

A255614 G.f.: Product_{k>=1} 1/(1-x^k)^(7*k).

Original entry on oeis.org

1, 7, 42, 203, 882, 3486, 12880, 44885, 149170, 475587, 1462993, 4359474, 12628091, 35656446, 98372109, 265701212, 703800790, 1830960824, 4684293222, 11798774953, 29288385021, 71714795158, 173351031721, 413964243476, 977243358574, 2281942600035, 5273570826594

Offset: 0

Vaclav Kotesovec, Feb 28 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
Eric Weisstein's World of Mathematics, Plane Partition
Wikipedia, Plane partition

Cf. A000219, A161870, A255610, A255611, A255612, A255613, A193427.

Column k=7 of A255961.

a:= proc(n) option remember; `if`(n=0, 1, 7*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(7*k),{k,1,nmax}],{x,0,nmax}],x]

G.f.: Product_{k>=1} 1/(1-x^k)^(7*k).
a(n) ~ 7^(13/36) * Zeta(3)^(13/36) * exp(7/12 + 3 * 2^(-2/3) * 7^(1/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^7 * 2^(5/36) * sqrt(3*Pi) * n^(31/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015
G.f.: exp(7*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

New name from Vaclav Kotesovec, Mar 12 2015

A276554 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -2, 0, 1, -3, -3, -1, 0, 1, -4, -3, 2, 0, 0, 1, -5, -2, 8, 6, 4, 0, 1, -6, 0, 16, 12, 12, 4, 0, 1, -7, 3, 25, 13, 9, 1, 7, 0, 1, -8, 7, 34, 5, -12, -29, -10, 3, 0, 1, -9, 12, 42, -15, -51, -78, -54, -32, -2, 0, 1, -10, 18, 48, -49, -102

Offset: 0

Seiichi Manyama, Apr 10 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -2, -3, -3, -2, ...
   0, -1,  2,  8, 16, ...
   0,  0,  6, 12, 13, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A000007, A073592, A276551, A276552, A316463, A316464.
Main diagonal gives A281267.
Antidiagonal sums give A299211.
Cf. A255961, A277938.

G.f. of column k: Product_{j>=1} (1-x^j)^(j*k).

A279928 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, -2, 0, 1, -4, 0, -2, 1, 0, 1, -5, 2, -1, 7, 0, 0, 1, -6, 5, 0, 15, 2, 4, 0, 1, -7, 9, 0, 23, -3, 10, 2, 0, 1, -8, 14, -2, 30, -20, 8, -8, 8, 0, 1, -9, 20, -7, 36, -51, 2, -42, 5, -2, 0, 1, -10, 27, -16, 42, -96, 5, -88, 6

Offset: 0

Seiichi Manyama, Apr 11 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -1, -1,  0,  2, ...
   0, -2, -2, -1,  0, ...
   0,  1,  7, 15, 23, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A000007, A255528, A278710, A279031, A279411, A279932.
Main diagonal gives A281266.
Antidiagonal sums give A299212.
Cf. A255961, A277938.

G.f. of column k: Product_{j>=1} 1/(1+x^j)^(j*k).

A101853 a(n) = n(20 + 15n + n^2)/6.

Original entry on oeis.org

6, 18, 37, 64, 100, 146, 203, 272, 354, 450, 561, 688, 832, 994, 1175, 1376, 1598, 1842, 2109, 2400, 2716, 3058, 3427, 3824, 4250, 4706, 5193, 5712, 6264, 6850, 7471, 8128, 8822, 9554, 10325, 11136, 11988, 12882, 13819, 14800

Offset: 1

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 18 2004

4th partial summation within series as series accumulate n times from an initial sequence of Euler Triangle's row 3: 1,4,1. The 1,4,1 is the left column, A101101 the second column, A008458 the third, A003215 the fourth column etc of the array in the example. a(n) is the 4th row.

			Left column the third row of A008292, and subsequent columns defined as partial sums along their preceding neighbors:
1 1  1   1   1    1    1     1     1     1     1
4 5  6   7   8    9   10    11    12    13    14
1 6 12  19  27   36   46    57    69    82    96  A051936
0 6 18  37  64  100  146   203   272   354   450  A101853
0 6 24  61 125  225  371   574   846  1200  1650  A101854
0 6 30  91 216  441  812  1386  2232  3432  5082  A101855
0 6 36 127 343  784 1596  2982  5214  8646 13728
0 6 42 169 512 1296 2892  5874 11088 19734 33462
0 6 48 217 729 2025 4917 10791 21879 41613 75075
...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Row n=3 of A255961.

Cf. A003215, A008292, A008458, A101101.

I:=[6, 18, 37, 64]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 26 2012

LinearRecurrence[{4,-6,4,-1},{6,18,37,64},40] (* or *) CoefficientList[Series[(6-6*x+x^2)/(x-1)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 26 2012 *)

a(n)=n*(20+15*n+n^2)/6 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: x*(6 - 6*x + x^2)/(x - 1)^4. - R. J. Mathar, Dec 06 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 26 2012
E.g.f.: exp(x)*x*(36 + 18*x + x^2)/6. - Stefano Spezia, Oct 14 2022

A316461 Expansion of Product_{k>=1} 1/(1-x^k)^(9*k).

Original entry on oeis.org

1, 9, 63, 354, 1755, 7866, 32682, 127458, 471681, 1668313, 5673501, 18635229, 59342520, 183768255, 554843493, 1636855647, 4727195028, 13386032649, 37219413972, 101741003451, 273721954086, 725498278359, 1896086574252, 4890111992460, 12454590256587

Offset: 0

Seiichi Manyama, Jul 04 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Column k=9 of A255961.

A316462 Expansion of Product_{k>=1} 1/(1-x^k)^(10*k).

Original entry on oeis.org

1, 10, 75, 450, 2365, 11202, 49020, 200900, 779650, 2886970, 10263057, 35193340, 116865525, 376994900, 1184596410, 3633899268, 10904012980, 32058245900, 92484729375, 262142323480, 730865120997, 2006375900880, 5428232646875, 14485462366250, 38155463010810

Offset: 0

Seiichi Manyama, Jul 04 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Column k=10 of A255961.

cp's OEIS Frontend

A255613 G.f.: Product_{k>=1} 1/(1-x^k)^(6*k).

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A255614 G.f.: Product_{k>=1} 1/(1-x^k)^(7*k).

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A276554 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-x^j)^(j*k) in powers of x.

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A279928 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j*k) in powers of x.

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A101853 a(n) = n*(20 + 15*n + n^2)/6.

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A316461 Expansion of Product_{k>=1} 1/(1-x^k)^(9*k).

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A316462 Expansion of Product_{k>=1} 1/(1-x^k)^(10*k).

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A101853 a(n) = n(20 + 15n + n^2)/6.