A117487 -id:A117487 - cp's OEIS frontend

A117486 Expansion of 1/((1-x)(1-x^2)(1-x^3)*(1-x^4))^2.

1, 2, 5, 10, 20, 34, 59, 94, 149, 224, 334, 480, 685, 950, 1307, 1762, 2357, 3100, 4050, 5220, 6685, 8466, 10659, 13294, 16494, 20298, 24859, 30234, 36609, 44056, 52806, 62952, 74770, 88380, 104112, 122116, 142786, 166304, 193134, 223504, 257954, 296756, 340544

Offset: 0

Alford Arnold, Mar 22 2006

Molien series for S_4 X S_4, cf. A001400.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1,-4,4,4,2,0,-10,0,2,4,4,-4,-1,-2,1,2,-1).
Index entries for two-way infinite sequences

Column four of table A115994.
Cf. A001400, A001399, A117485, A117487.
Cf. A000027, A006918, A115994.

n:=4; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H);

CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3)(1-x^4))^2,{x,0,50}],x] (* Harvey P. Dale, Jul 22 2012 *)

a(n):=floor(2*floor(-n/3)*cos(2*%pi*(n+1)/3)/81+(n+2)*cos(%pi*n/ 2)/128+(n+1)*(2835*(n^2+29*n+246)*(-1)^n+6*n^6+414*n^5+11556*n^4 +166944*n^3 +1320045*n^2+5489625*n+10008110)/ 17418240+1/2); /* Tani Akinari, Nov 14 2012 */

Vec(1 / ((1 - x)^8*(1 + x)^4*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^40)) \\ Colin Barker, Apr 07 2019

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))^2.

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) - 4*a(n-5) + 4*a(n-6) + 4*a(n-7) + 2*a(n-8) - 10*a(n-10) + 2*a(n-12) + 4*a(n-13) + 4*a(n-14) - 4*a(n-15) - a(n-16) - 2*a(n-17) + a(n-18) + 2*a(n-19) - a(n-20) for n>19. - Colin Barker, Apr 07 2019

Entry revised by N. J. A. Sloane, Mar 10 2007

A330643 a(n) is the number of partitions of n with Durfee square of size <= 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17976, 21635, 26010, 31175, 37318, 44547, 53109, 63153, 74996, 88850, 105113, 124078, 146256, 172032, 202056, 236844

Offset: 0

Omar E. Pol, Dec 24 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -2, -1, -2, 0, 2, 6, 2, -3, -6, -5, -2, 3, 12, 3, -2, -5, -6, -3, 2, 6, 2, 0, -2, -1, -2, 1, 2, -1).

Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A117486, A117487, A330640, A330641, A330642.

seq(n) = Vec(sum(k=0, 5, x^(k^2)/prod(j=1, k, 1 - x^j)^2) + O(x*x^n)) \\ Andrew Howroyd, Dec 27 2024

a(n) = A000041(n), 0 <= n <= 35.
a(n) = A330642(n), 0 <= n <= 24.
a(n) = A330642(n) + A117487(n-24), n >= 25.
a(n) = n + A006918(n-3) + A117485(n) + A117486(n-16) + A117487(n-24), n >= 25.
G.f.: Sum_{k=0..5} x^(k^2)/(Product_{j=1..k} (1 - x^j))^2. - Andrew Howroyd, Dec 27 2024

A160647 Self-convolution of sequence A001402.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 108, 179, 284, 445, 676, 1017, 1492, 2168, 3094, 4372, 6088, 8406, 11462, 15509, 20770, 27614, 36390, 47646, 61898, 79939, 102538, 130808, 165864, 209272, 262598, 328008, 407700, 504607, 621760, 763123, 932788, 1136047

Offset: 1

Alford Arnold, May 27 2009

			a(8) = 108 because the eighth antidiagonal of the associated array is 14 11 14 15 15 14 11 14 and sums to 108.

Cf. A117566.

Sixth in a list of sequences related to numeric partitions; earlier sequences are A000027, A006918, A117485, A117486, and A117487.

A160647 := proc(n) coeftayl( convert(1/mul((1-x^j)^2,j=1..6),parfrac,x),x=0,n) ; end: seq(A160647(n),n=0..45) ; # R. J. Mathar, Jun 16 2009

More terms from R. J. Mathar, Jun 16 2009

A364842 Table read by antidiagonals: row n gives the Euler transform of the sequence (2,...,2,0,0,...) that contains n 2s followed by 0s.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 5, 8, 5, 1, 2, 5, 10, 14, 6, 1, 2, 5, 10, 18, 20, 7, 1, 2, 5, 10, 20, 30, 30, 8, 1, 2, 5, 10, 20, 34, 49, 40, 9, 1, 2, 5, 10, 20, 36, 59, 74, 55, 10, 1, 2, 5, 10, 20, 36, 63, 94, 110, 70, 11, 1, 2, 5, 10, 20, 36, 65, 104, 149, 158, 91, 12

Offset: 1

Peter Kagey, Nov 09 2023

			Table begins:
  | 0 1 2  3  4  5  6   7   8   9  10
--+----------------------------------
1 | 1 2 3  4  5  6  7   8   9  10  11
2 | 1 2 5  8 14 20 30  40  55  70  91
3 | 1 2 5 10 18 30 49  74 110 158 221
4 | 1 2 5 10 20 34 59  94 149 224 334
5 | 1 2 5 10 20 36 63 104 169 264 405
6 | 1 2 5 10 20 36 65 108 179 284 445
7 | 1 2 5 10 20 36 65 110 183 294 465
8 | 1 2 5 10 20 36 65 110 185 298 475
9 | 1 2 5 10 20 36 65 110 185 300 479

Cf. A000027 (row 1), A006918 (row 2), A117485 (row 3), A117486 ( row 4), A117487 (row 5), A160647 (row 6), A000712 (main diagonal).
Analogous for initial 1s sequence A008284.
Cf. A115994.

Seed[i_, n_] := ConstantArray[2, i]~Join~ConstantArray[0, n - i];
A364842Table[n_] := Table[Seed[i, n] // EulerTransform, {i, 1, n}]
(*EulerTransform is defined in A005195*)

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A117486 Expansion of 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))^2.

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A330643 a(n) is the number of partitions of n with Durfee square of size <= 5.

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A160647 Self-convolution of sequence A001402.

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A364842 Table read by antidiagonals: row n gives the Euler transform of the sequence (2,...,2,0,0,...) that contains n 2s followed by 0s.

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A117486 Expansion of 1/((1-x)(1-x^2)(1-x^3)*(1-x^4))^2.