A001401 -id:A001401 - cp's OEIS frontend

A002622 Number of partitions of at most n into at most 5 parts.

1, 2, 4, 7, 12, 19, 29, 42, 60, 83, 113, 150, 197, 254, 324, 408, 509, 628, 769, 933, 1125, 1346, 1601, 1892, 2225, 2602, 3029, 3509, 4049, 4652, 5326, 6074, 6905, 7823, 8837, 9952, 11178, 12520, 13989, 15591, 17338, 19236, 21298, 23531, 25949, 28560, 31378, 34412

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 29*x^6 + 42*x^7 + 60*x^8 + ...
a(2) = 4 with partitions 0, 1, 2, 1+1. a(3) = 7 with partitions 0, 1, 2, 1+1, 3, 2+1, 1+1+1. - _Michael Somos_, Apr 24 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 0, 2, -1).

Cf. A001401 (first differences). Column 5 of A092905.

CoefficientList[Series[1/((1 - x)^2 (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Apr 25 2014 *)
LinearRecurrence[{2, 0, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 0, 2, -1},  {1, 2, 4, 7, 12, 19, 29, 42, 60, 83, 113, 150, 197, 254, 324, 408},  48] (* Georg Fischer, Feb 27 2019 *)

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 5, 1-x^i))) \\ Altug Alkan, Mar 30 2018

G.f.: 1/[(1+x^2)*(1-x^3)*(1-x)^4*(1-x^5)*(1+x)^2]. (Corrected Mar 31 2018)
a(n)= 2*a(n-1) -a(n-3) -a(n-5) +2*a(n-8) -a(n-11) -a(n-13) +2*a(n-15) -a(n-16).
G.f.: 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^3) * (1 - x^4) * (1 - x^5)). - Michael Somos, Apr 24 2014
Euler transform of length 5 sequence [ 2, 1, 1, 1, 1]. - Michael Somos, Apr 24 2014
a(n) = a(n-1) + A001401(n). - Michael Somos, Apr 24 2014
a(n) = round((n+1)*(6*n^4+234*n^3+3326*n^2+20674*n+50651+675*(-1)^n)/86400). - Tani Akinari, May 05 2014

A103924 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4 and 5.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 64, 107, 177, 282, 443, 678, 1026, 1522, 2234, 3231, 4628, 6550, 9193, 12774, 17619, 24098, 32740, 44161, 59213, 78894, 104553, 137787, 180702, 235806, 306354, 396226, 510392, 654787, 836911, 1065734, 1352475, 1710535, 2156536, 2710318

Offset: 0

Wolfdieter Lang, Mar 24 2005

See A103923 for other combinatorial interpretations of a(n).
Convolution of A001401 and A000041. - Vaclav Kotesovec, Aug 28 2015
Also the sum of binomial (D(p), 5) over partitions p of n+15, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Sixth column (m=5) of Fine-Riordan triangle A008951 and of triangle A103923, i.e. the p_2(n, m) array of the Gupta et al. reference.

Cf. A000712 (all parts of two kinds).

with(numtheory): a:= proc(n) a(n):=`if`(n=0, 1, add(add(d*`if`(d<6, 2, 1), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Sep 14 2014

a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*If[d<6, 2, 1], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 28 2015, after Alois P. Heinz *)
nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 5}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@5],  {n,0,39}] (* Robert Price, Jul 29 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m(m+1)/2) := T[n, m] = T[n-m, m-1] + T[n-m, m];
T[, ] = 0;
a[n_] := T[n+15, 5];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

G.f.: (product(1/(1-x^k), k=1..5)^2)*product(1/(1-x^j), j=6..infty).
a(n) = sum(A000710(n-5*j), j=0..floor(n/5)), n>=0.
a(n) ~ 3*n^(3/2) * exp(Pi*sqrt(2*n/3)) / (20*sqrt(2)*Pi^5). - Vaclav Kotesovec, Aug 28 2015

A085756 Number of partitions into a prime number of distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 20, 22, 26, 30, 35, 40, 47, 53, 62, 71, 82, 93, 108, 123, 141, 161, 184, 209, 239, 271, 309, 350, 397, 449, 509, 575, 649, 732, 825, 928, 1044, 1172, 1315, 1474, 1650, 1845, 2061, 2300, 2563, 2854, 3174, 3526, 3912, 4337

Offset: 3

Vladeta Jovovic, Jul 21 2003

nonn

			a(15)=20 because there are 20 partitions of 15 into a prime number of distinct parts: 1+2+3+4+5=4+5+6=3+5+7=2+6+7=3+4+8=2+5+8=1+6+8=7+8=2+4+9=1+5+9=6+9=2+3+10=
1+4+10=5+10=1+3+11=4+11=1+2+12=3+12=2+13=1+14.

Cf. A038499.

a(n) = A004526(n-1) +A001399(n-6) +A001401(n-15) +A008636(n-28) + .... - R. J. Mathar, Feb 13 2019

A341912 Number of partitions of n into 5 distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 83, 101, 118, 141, 162, 192, 218, 255, 286, 333, 370, 427, 470, 540, 590, 673, 730, 831, 894, 1014, 1085, 1224, 1305, 1469, 1552, 1747, 1841, 2057, 2163, 2418, 2520, 2818, 2933, 3256, 3388, 3765, 3879, 4319, 4452, 4914, 5068

Offset: 15

Ilya Gutkovskiy, Feb 23 2021

nonn

Cf. A000743, A023022, A023025, A078374, A101271, A339672, A340719, A341868, A341870, A341913, A341914.

nmax = 70; CoefficientList[Series[Sum[MoebiusMu[k] x^(15 k)/Product[1 - x^(j k), {j, 1, 5}], {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 15] &

G.f.: Sum_{k>=1} mu(k)* x^(15*k) / Product_{j=1..5} (1 - x^(j*k)).

a(n) <= A001401(n-15). - R. J. Mathar, Feb 28 2021

A117487 G.f.: 1/((1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5))^2.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 63, 104, 169, 264, 405, 604, 888, 1278, 1815, 2536, 3502, 4772, 6437, 8586, 11352, 14866, 19315, 24890, 31851, 40466, 51089, 64092, 79952, 99172, 122386, 150264, 183639, 223394, 270605, 326422, 392225, 469490, 559970, 665542, 788412

Offset: 1

Alford Arnold, Mar 22 2006

nonn

Molien series for S_5 X S_5, cf. A001401.
Molien series for S_k X S_k approaches A000712 as k increases.
Column 5 of table A115994.
Note that a(5) is 20, the scalar product of (1 1 2 3 5) and (5 3 2 1 1 ). a(6) is 36, the scalar product of (1 1 2 3 5 7) and (7 5 3 2 1 1 ).

G. C. Greubel, Table of n, a(n) for n = 1..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. A000027, A000712, A001399, A001400, A006918.

Cf. A115994, A117485, A117486, A117488, A117489.

n:=5; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // N. J. A. Sloane

# adapted from A115994 kmax := 120 : qmax := kmax/2 : g:=sum(t^k*q^(k^2)/product((1-q^j)^2, j=1..k), k=1..kmax): gser:=series(g, q=0, qmax): for n from 25 to qmax-1 do P :=coeff(gser, q^n) : printf("%a,",coeff(P, t^5)); od: # R. J. Mathar, Apr 07 2006

CoefficientList[Series[1/(Product[(1-x^j), {j,5}])^2, {x,0,45}], x] (* G. C. Greubel, Jan 01 2020 *)

my(x='x+O('x^45)); Vec( 1/(prod(j=1,5, 1-x^j))^2 ) \\ G. C. Greubel, Jan 01 2020

def A117487_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(product(1-x^j for j in (1..5)))^2 ).list()
A117487_list(45) # G. C. Greubel, Jan 01 2020

More terms from R. J. Mathar, Apr 07 2006

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

A145574 Array a(n,m) for number of partitions of n>=2 with m parts having no part 1. Hence m=1..floor(n/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 3, 1, 1, 4, 4, 2, 1, 1, 4, 5, 3, 1, 1, 5, 7, 5, 2, 1, 1, 5, 8, 6, 3, 1, 1, 6, 10, 9, 5, 2, 1, 1, 6, 12, 11, 7, 3, 1, 1, 7, 14, 15, 10, 5, 2, 1, 1, 7, 16, 18, 13, 7, 3, 1, 1, 8, 19, 23, 18, 11, 5, 2, 1, 1, 8, 21, 27, 23, 14, 7, 3, 1, 1, 9, 24, 34, 30

Offset: 2

Wolfdieter Lang and Malin Sjodahl, Mar 06 2009

The row lengths sequence is floor(n/2) = [1,1,2,2,3,3,4,4,...], see A008619(n-1), n>=2.
Obtained from the characteristic partition array A145573 by summing in row n>=2 over entries belonging to like parts number m.
The column sequences give A000012, A004526, A001399, A001400, A001401, A001402, A026813 for m=1..7.

			1;
1;
1, 1;
1, 1;
1, 2, 1;
1, 2, 1;
1, 3, 2, 1;
1, 3, 3, 1;
1, 4, 4, 2, 1;

Alois P. Heinz, Rows n = 2..200, flattened
W. Lang and M. Sjodahl, First 20 rows of the array and row sums.

Cf. A145573, A002865 (row sums).

b:= proc(n, i, t) option remember; `if`(2*t>n or t*i b(n, n, m):
seq(seq(a(n, m), m=1..iquo(n, 2)), n=2..30); # Alois P. Heinz, Oct 18 2012

nn=15; f[list_]:=Select[list,#>0&]; p=Product[1/(1-y x^i), {i, 2, nn}]; Drop[Map[f, CoefficientList[Series[p, {x, 0, nn}], {x, y}]], 1]//Grid  (* Geoffrey Critzer, Sep 23 2012 *)

# Prints the table; cf. A011973.
for n in (2..20): [Partitions(n, length=m, min_part=2).cardinality() for m in (1..n//2)]  # Peter Luschny, Oct 18 2012

a(n,m) = sum over entries of A145573(n,k) array which belong to partitions with part number m, for m=1..floor(n/2)). Note that partitions with parts number m>floor(n/2) have always at least one part 1.

G.f.: Product_{i>=2} 1/(1- y*x^i). - Geoffrey Critzer, Sep 23 2012

A256225 Number of partitions of 5n into 5 parts.

Original entry on oeis.org

0, 1, 7, 30, 84, 192, 377, 674, 1115, 1747, 2611, 3765, 5260, 7166, 9542, 12470, 16019, 20282, 25337, 31289, 38225, 46262, 55496, 66055, 78045, 91606, 106852, 123935, 142979, 164147, 187572, 213429, 241860, 273052, 307156, 344370, 384855, 428821, 476437, 527925

Offset: 0

Colin Barker, Mar 19 2015

			For n=2, the 7 partitions of 10 are [6,1,1,1,1], [5,2,1,1,1], [4,3,1,1,1], [4,2,2,1,1], [3,3,2,1,1], [3,2,2,2,1] and [2,2,2,2,2].

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

Cf. A001401, A077043, A238340, A256226, A256235.

Length /@ (Length /@ IntegerPartitions[5 #, {5}] & /@ Range@ 39) (* Michael De Vlieger, Mar 20 2015 *)

concat(0, Vec(-x* (x^8+5*x^7+16*x^6+25*x^5+31*x^4+25*x^3+16*x^2+5*x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^2+x+1)) + O(x^100)))

concat(0, vector(40, n, k=0; forpart(p=5*n, k++, , [5,5]); k)) \\ Colin Barker, Mar 21 2015

G.f.: -x*(x^8+5*x^7+16*x^6+25*x^5+31*x^4+25*x^3+16*x^2+5*x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^2+x+1)).

A091492 Triangle, read by rows, generated recursively and related to partitions.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 1, 1, 3, 1, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 0, 0, 0, 1, 1, 4, 3, 0, 0, 0, 0, 0, 0, 1, 1, 4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 1, 5, 5, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 5, 7, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 8, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Jan 16 2004

Excluding the leading zeros, the columns are related to partitions. The 3rd column lists A001399 (partitions of n into at most 3 parts). The 4th column lists A001400 (partitions of n into at most 4 parts). The 5th column lists A001401 (partitions of n into at most 5 parts). The 6th column is A091498. Row sums are A091493. The number of nonzero terms in each row is A091497.

			T(12,3) = 7 = (4)*1+(3)*1 = T(9,2)*T(2,1)+T(9,3)*T(3,0) = Sum T(9,j)*T(j,3-j) {j=2..3}.
Rows begin:
{1},
{1,1},
{1,1,0},
{1,1,1,0},
{1,1,1,0,0},
{1,1,2,0,0,0},
{1,1,2,1,0,0,0},
{1,1,3,1,0,0,0,0},
{1,1,3,2,0,0,0,0,0},
{1,1,4,3,0,0,0,0,0,0},
{1,1,4,4,1,0,0,0,0,0,0},
{1,1,5,5,1,1,0,0,0,0,0,0},
{1,1,5,7,2,1,0,0,0,0,0,0,0},
{1,1,6,8,3,2,0,0,0,0,0,0,0,0},
{1,1,6,10,5,3,0,0,0,0,0,...
{1,1,7,12,6,5,0,0,0,0,0,...
{1,1,7,14,9,7,1,0,0,0,0,...
{1,1,8,16,11,10,2,0,0,0,...
{1,1,8,19,15,13,3,2,0,0,...
{1,1,9,21,18,18,5,2,0,0,...
{1,1,9,24,23,23,8,4,0,0,...
{1,1,10,27,27,30,11,6,0,...
{1,1,10,30,34,37,17,10,0,...

Cf. A001399, A001400, A001401, A091493, A091497, A091498.

T(n,k)=if(k>n || n<0 || k<0,0,if(k<=1 || (k==n && n<2),1, sum(j=(k+1)\2,min(n-k,k),T(n-k,j)*T(j,k-j)););)

T(n, k)=Sum T(n-k, j)*T(j, k-j) {j=[(k+1)/2]..min(k, n-k)}, with T(0, 0)=1, T(n, 0)=1, T(1, 1)=1.

A139672 Convolution of A008619 and A001400.

Original entry on oeis.org

1, 2, 5, 9, 17, 27, 44, 65, 97, 136, 191, 257, 346, 451, 587, 746, 946, 1177, 1461, 1786, 2178, 2623, 3151, 3746, 4443, 5223, 6126, 7131, 8283, 9558, 11007, 12603, 14403, 16377, 18588, 21003, 23692, 26618, 29858, 33372, 37244, 41430, 46022, 50972

Offset: 1

Alford Arnold, Apr 29 2008, May 01 2008

nonn

This is row 21 of a table of values related to Molien series. It is the product of the sequence on row 3 (A008619) with the sequence on row 7 (A001400).
This table may be constructed by moving the rows of table A008284 to prime locations and generating the composite locations by multiplication in a manner similar to the calculation illustrated in the present sequence.
Rows 1 thru 20 and 22 thru 25 are as follows:
A000012 A008619 A000027 A001399 A002620 A001400 A000217 A006918 A000601 A001401 A002623 A001402 A002621 A097701 A000292 A008630 A002624 A008631 A057524 A002622 A008632 A001752 A117485.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -3, 0, -1, 2, 2, -1, 0, -3, 1, 2, -1).

a:= proc(n) local m, r; m:= iquo (n, 12, 'r'); r:= r+1; (19+ (145+ (260+ 15* (r+9)*r+ (405+ 90*r+ 216*m) *m) *m) *m) *m/5+ [0, 1, 2, 5, 9, 17, 27, 44, 65, 97, 136, 191][r]+ [0, 16, 37, 77, 128, 208, 307, 447, 616, 840, 1105, 1441][r]*m/2+ [0, 52, 119, 213, 328, 476, 651, 865, 1112, 1404, 1735, 2117][r]*m^2/2 end: seq (a(n), n=1..50); # Alois P. Heinz, Nov 10 2008

CoefficientList[Series[x/((x^2+x+1)(x^2+1)(x+1)^3 (x-1)^6),{x,0,50}],x] (* or *) LinearRecurrence[{2,1,-3,0,-1,2,2,-1,0,-3,1,2,-1},{0,1,2,5,9,17,27,44,65,97,136,191,257},50] (* Harvey P. Dale, Feb 17 2016 *)

G.f.: x/((x^2+x+1)*(x^2+1)*(x+1)^3*(x-1)^6). - Alois P. Heinz, Nov 10 2008

a(n)= -A049347(n)/27 +(2*n+11)*(6*n^4+132*n^3+914*n^2+2068*n+1055)/69120 -(-1)^n*(51/512+n^2/256+11*n/256+A057077(n)/32 ). - R. J. Mathar, Nov 21 2008

More terms from Alois P. Heinz, Nov 10 2008

Corrected A-number in definition. Added formula. - R. J. Mathar, Nov 21 2008

A288166 Expansion of x^5/((1-x^5)(1-x^4)(1-x^8)(1-x^12)(1-x^16)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 3, 2, 1, 1, 5, 3, 2, 1, 7, 5, 3, 2, 10, 7, 5, 3, 13, 10, 7, 5, 18, 13, 10, 7, 23, 18, 13, 10, 30, 23, 18, 13, 37, 30, 23, 18, 47, 37, 30, 23, 57, 47, 37, 30, 70, 57, 47, 37, 84, 70, 57, 47, 101, 84, 70, 57, 119

Offset: 0

Seiichi Manyama, Jun 06 2017

			a(56) = p_5(56/4)      = p_5(14) = A001401(9)  = 23,
a(57) = p_5((57+15)/4) = p_5(18) = A001401(13) = 57,
a(58) = p_5((58+10)/4) = p_5(17) = A001401(12) = 47,
a(59) = p_5((59+5)/4)  = p_5(16) = A001401(11) = 37,
a(60) = p_5(60/4)      = p_5(15) = A001401(10) = 30,
a(61) = p_5((61+15)/4) = p_5(19) = A001401(14) = 70,
a(62) = p_5((62+10)/4) = p_5(18) = A001401(13) = 57,
a(63) = p_5((63+5)/4)  = p_5(17) = A001401(12) = 47.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Daniel Panario, Murat Sahin and Qiang Wang, Generalized Alcuin’s Sequence, The Electronic Journal of Combinatorics, Volume 19, Issue 4 (2012).
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 1, 1, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, -1, -1, 0, 0, 0, 1).

Cf. A001401.

Cf. A005044 (k=3), A288165 (k=4), this sequence (k=5).

CoefficientList[Series[x^5/((1-x^4)(1-x^5)(1-x^8)(1-x^12)(1-x^16)),{x,0,120}],x] (* or *) LinearRecurrence[ {0,0,0,1,1,0,0,1,-1,0,0,0,-1,0,0,0,0,0,0,-2,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,1,-1,0,0,-1,-1,0,0,0,1},{0,0,0,0,0,1,0,0,0,1,1,0,0,2,1,1,0,3,2,1,1,5,3,2,1,7,5,3,2,10,7,5,3,13,10,7,5,18,13,10,7,23,18,13,10},120] (* Harvey P. Dale, Apr 22 2019 *)

a(n) = p_5(n/4)      if n == 0 mod 4,
a(n) = p_5((n+15)/4) if n == 1 mod 4,
a(n) = p_5((n+10)/4) if n == 2 mod 4,
a(n) = p_5((n+5)/4)  if n == 3 mod 4,
where p_5(n) is the number of partitions of n into exactly 5 parts.

cp's OEIS Frontend

A002622 Number of partitions of at most n into at most 5 parts.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A103924 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4 and 5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A085756 Number of partitions into a prime number of distinct parts.

Views

Author

Keywords

Examples

Crossrefs

Formula

A341912 Number of partitions of n into 5 distinct and relatively prime parts.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A117487 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Extensions

A145574 Array a(n,m) for number of partitions of n>=2 with m parts having no part 1. Hence m=1..floor(n/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A256225 Number of partitions of 5n into 5 parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A091492 Triangle, read by rows, generated recursively and related to partitions.

A117487 G.f.: 1/((1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5))^2.

A288166 Expansion of x^5/((1-x^5)(1-x^4)(1-x^8)(1-x^12)(1-x^16)).