A026812 -id:A026812 - cp's OEIS frontend

A060025 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 6.

1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 2, 3, -1, -1, -6, -9, -17, -22, -35, -43, -61, -76, -100, -121, -155, -185, -229, -271, -328, -383, -458, -529, -622, -715, -830, -946, -1090, -1233, -1407, -1584, -1794, -2008, -2261, -2517, -2816, -3124, -3476, -3838, -4253, -4677, -5159, -5656, -6213

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+5 into 5 parts and the number of partitions of n+5 into 6 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Cf. A026811, A026812.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), this sequence (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

m:=6; R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x-x^m)/( (&*[1-x^j: j in [1..m]]) ) )); // G. C. Greubel, Apr 17 2019

With[{nn=6},CoefficientList[Series[(1-x-x^nn)/Times@@(1-x^Range[nn]),{x,0,60}],x]] (* Harvey P. Dale, May 15 2016 *)

Vec((1 - x - x^6) / ((1 - x)^6*(1 + x)^3*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Apr 17 2019

m=6; ((1-x-x^m)/( product(1-x^j for j in (1..m)) )).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Apr 17 2019

a(n) = A026811(n+5) - A026812(n+5). - Wesley Ivan Hurt, Apr 16 2019

G.f.: (1 - x - x^6) / ((1 - x)^6*(1 + x)^3*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Apr 17 2019

A060026 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 7.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 5, 5, 7, 6, 9, 6, 8, 5, 5, -1, -2, -13, -18, -33, -45, -68, -86, -121, -151, -198, -244, -310, -373, -464, -553, -671, -793, -948, -1107, -1309, -1517, -1771, -2039, -2360, -2696, -3098, -3519, -4011, -4534, -5137, -5774, -6508, -7283, -8163, -9099, -10153, -11269

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+6 into 6 parts and the number of partitions of n+6 into 7 parts. - Wesley Ivan Hurt, Apr 16 2019

Ray Chandler, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, -1, 0, 1, 1, 2, 0, 0, 0, -2, -1, -1, 0, 1, 1, 0, 1, 0, 0, -1, -1, 1).

Cf. A026812, A026813.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), this sequence (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

With[{nn=7},CoefficientList[Series[(1-x-x^nn)/Times@@(1-x^Range[nn]),{x,0,60}],x]] (* Harvey P. Dale, May 15 2016 *)

a(n) = A026812(n+6) - A026813(n+6). - Wesley Ivan Hurt, Apr 16 2019

A211861 Number of partitions of n into parts <= 6 with the property that all parts have distinct multiplicities.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 9, 12, 13, 18, 23, 25, 36, 43, 45, 60, 75, 78, 102, 108, 126, 151, 184, 188, 237, 260, 305, 339, 408, 415, 521, 548, 627, 689, 815, 824, 997, 1050, 1202, 1287, 1497, 1537, 1831, 1903, 2166, 2288, 2658, 2721, 3156, 3274

Offset: 0

Matthew C. Russell, Apr 25 2012

nonn

			For n=3 the a(3)=2 partitions are {3} and {1,1,1}. Note that {2,1} does not count, as 1 and 2 appear with the same nonzero multiplicity.

Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions.

Cf. A026812, A098859.

Cf. A105637, A211858, A211859, A211860, A211862, A211863.

a211861 n = p 0 [] [1..6] n where
   p m ms _      0 = if m `elem` ms then 0 else 1
   p    []     _ = 0
   p m ms ks'@(k:ks) x
     | x < k       = 0
     | m == 0      = p 1 ms ks' (x - k) + p 0 ms ks x
     | m `elem` ms = p (m + 1) ms ks' (x - k)
     | otherwise   = p (m + 1) ms ks' (x - k) + p 0 (m : ms) ks x
-- Reinhard Zumkeller, Dec 27 2012

A008641 Number of partitions of n into at most 12 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 172, 224, 285, 366, 460, 582, 725, 905, 1116, 1380, 1686, 2063, 2503, 3036, 3655, 4401, 5262, 6290, 7476, 8877, 10489, 12384, 14552, 17084, 19978, 23334, 27156, 31570, 36578, 42333, 48849, 56297

Offset: 0

N. J. A. Sloane

With a different offset, number of partitions of n in which the greatest part is 12.

Also number of partitions of n into parts <= 12: a(n)=A026820(n,12). [Reinhard Zumkeller, Jan 21 2010]

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 361
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, -1, 0, 2, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, -2, 0, 1, 2, 2, 2, 2, 1, 1, 0, -1, -2, -1, -4, -1, -2, -1, 0, 1, 1, 2, 2, 2, 2, 1, 0, -2, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 2, 0, -1, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

a(n) = A008284(n+12, 12), n >= 0.

Cf. A026810, A026811, A026812, A026813, A026814, A026815, A026816.

1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)/(1-x^9)/(1-x^10)/(1-x^11)/(1-x^12)
with(combstruct):ZL13:=[S,{S=Set(Cycle(Z,card<13))}, unlabeled]:seq(count(ZL13,size=n),n=0..46); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=12)},unlabelled]: seq(combstruct[count](B, size=n), n=0..46); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 12} ], {x, 0, 60} ], x ]
Table[ Length[ Select[ Partitions[n], First[ # ] == 12 & ]], {n, 1, 60} ]

G.f.: 1/Product_{k=1..12}(1-x^k).

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) - a(n-13) + 2*a(n-15) + a(n-16) + a(n-17) - a(n-20) - a(n-21) - 2*a(n-22) - a(n-23) - a(n-24) - 2*a(n-26) + a(n-28) + 2*a(n-29) + 2*a(n-30) + 2*a(n-31) + 2*a(n-32) + a(n-33) + a(n-34) - a(n-36) - 2*a(n-37) - a(n-38) - 4*a(n-39) - a(n-40) - 2*a(n-41) - a(n-42) + a(n-44) + a(n-45) + 2*a(n-46) + 2a(n-47) + 2*a(n-48) + 2*a(n-49) + a(n-50) - 2*a(n-52) - a(n-54) - a(n-55) - 2*a(n-56) - a(n-57) - a(n-58) + a(n-61) + a(n-62) + 2*a(n-63) - a(n-65) + a(n-66) - a(n-71) - a(n-73) + a(n-76) + a(n-77) - a(n-78). - David Neil McGrath, Jul 28 2015

More terms from Robert G. Wilson v, Dec 11 2000

A234666 Number of combinations for the sum of 6 different numbers from 1 to 49.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5426, 5940, 6506, 7097, 7748, 8423

Offset: 21

Jacques ALARDET, Dec 29 2013

			a(21)=1 because the only way to sum to 21 is 1+2+3+4+5+6=21; a(279)=1 because 279 occurs only as 49+48+47+46+45+44.

Jacques ALARDET, Table of n, a(n) for n = 21..279

A026812 contains the same values until A026812(49)=4935.

A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 4, 3, 0, 1, 4, 4, 1, 0, 1, 5, 5, 1, 0, 1, 5, 7, 2, 0, 1, 6, 8, 3, 0, 1, 6, 10, 5, 0, 1, 7, 12, 6, 1, 0, 1, 7, 14, 9, 1, 0, 1, 8, 16, 11, 2, 0, 1, 8, 19, 15, 3, 0, 1, 9, 21, 18, 5, 0, 1, 9, 24, 23, 7

Offset: 0

Seiichi Manyama, Apr 07 2025

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2,  1;
  0, 1, 3,  1;
  0, 1, 3,  2;
  0, 1, 4,  3;
  0, 1, 4,  4, 1;
  0, 1, 5,  5, 1;
  0, 1, 5,  7, 2;
  0, 1, 6,  8, 3;
  0, 1, 6, 10, 5;
  0, 1, 7, 12, 6, 1;
  ...

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A000009.
Columns 0..10 give A000007, A000012, A004526(n-1), A069905(n-3), A026810(n-6), A026811(n-10), A026812(n-15), A026813(n-21), A026814(n-28), A026815(n-36), A026816(n-45).
Cf. A003056, A008284, A291954, A291960, A291968, A292047, A292049.

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1-x^j).

T(n,k) = |A292047(n,k)| = |A292049(n,k)|.

cp's OEIS Frontend

A060025 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 6.

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A060026 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 7.

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A211861 Number of partitions of n into parts <= 6 with the property that all parts have distinct multiplicities.

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A008641 Number of partitions of n into at most 12 parts.

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A234666 Number of combinations for the sum of 6 different numbers from 1 to 49.

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A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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