A025795 -id:A025795 - cp's OEIS frontend

A008667 Expansion of g.f.: 1/((1-x^2)(1-x^3)(1-x^4)*(1-x^5)).

1, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 22, 23, 28, 29, 34, 36, 42, 44, 50, 53, 60, 63, 71, 74, 83, 87, 96, 101, 111, 116, 127, 133, 145, 151, 164, 171, 185, 193, 207, 216, 232, 241, 258, 268, 286, 297, 316, 328, 348, 361, 382, 396, 419, 433, 457

Offset: 0

N. J. A. Sloane

Also, Molien series for invariants of finite Coxeter group A_4. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k not the alternating group Alt_k. - N. J. A. Sloane, Jan 11 2016

Number of partitions into parts 2, 3, 4, and 5. - Joerg Arndt, Apr 29 2014

			a(4)=2 because f''''(x)/4!=2 at x=0 for f=1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)).
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + 7*x^11 + ... .

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 32).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 241
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1,-2,-1,0,1,1,1,0,-1).

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

Cf. A005044, A001401 (partial sums).

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019

seq(coeff(series(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 08 2019

SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)),{x,0,#}]&/@Range[0,100] (* or *) a[k_]=SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4) (1-x^5)),{x,0,k}] (* Peter Pein (petsie(AT)dordos.net), Sep 09 2006 *)
CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,5}],{x,0,70}],x] (* Harvey P. Dale, Feb 22 2018 *)

{a(n) = if( n<-13, -a(-14 - n), polcoeff( prod( k=2, 5, 1 / (1 - x^k), 1 + x * O(x^n)), n))} /* Michael Somos, Oct 14 2006 */

def A008667_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))).list()
A008667_list(65) # G. C. Greubel, Sep 08 2019

Euler transform of length 5 sequence [ 0, 1, 1, 1, 1]. - Michael Somos, Sep 23 2006
a(-14 - n) = -a(n). - Michael Somos, Sep 23 2006
a(n) ~ 1/720*n^3. - Ralf Stephan, Apr 29 2014
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6) - 2*a(n-7) - a(n-8) + a(n-10) + a(n-11) + a(n-12) - a(n-14). - David Neil McGrath, Sep 13 2014
From R. J. Mathar, Jun 23 2021: (Start)
a(n)-a(n-2) = A008680(n).
a(n)-a(n-3) = A025802(n).
a(n)-a(n-4) = A025795(n).
a(n)-a(n-5) = A005044(n+3). (End)
a(n)= floor((n^3 + 21*n^2 + 156*n - 45*n*(n mod 2) + 720)/720 - [(n mod 10)=1]/5). - Hoang Xuan Thanh, Aug 20 2025

A078495 a(n) = (a(n-1) * a(n-6) + a(n-3) * a(n-4)) / a(n-7) (a variant of Somos-7).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 6, 12, 24, 72, 144, 288, 864, 3456, 10368, 41472, 124416, 497664, 2985984, 17915904, 71663616, 429981696, 2579890176, 20639121408, 185752092672, 1486016741376, 8916100448256, 106993205379072

Offset: 0

Michael Somos, Nov 26 2002

From Vladimir Shevelev, Apr 16 2016: (Start)
For k >= 0, an infinite sequence {b(k,n)} of Somos's sequences (n >= 0) is:
b(k,0) = b(k,1) = ... = b(k,2*k+2) = 1;
and then for n >= 2*k+3,
b(k,n) = (b(k,n-1)*b(k,n-2*k-2) + b(k,n-k-1)*b(k,n-k-2))/b(k,n-2*k-3).
In particular, {b(0,n)} is essentially A060656, {b(1,n)}=A006721, {a(2,n)}=A078495.
One can prove that the sequence {b(k,n)} has the first 4*(k+1) simple differences: 2k+2 zeros, after that k+1 1's and after that k+1 consecutive doubled triangular numbers (A000217), beginning with 2.
Further we have nontrivial differences. The first of them for k=0,1,2,... are 12, 26, 48, 80, 124, 182, 256, 348, 460, 594, ..., that is, {k^3/3 + 3*k^2 + 32*k/3 + 12}.
(End)

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Cf. A000217, A006721, A006723, A025795, A060656, A078495.

a078495 n = a078495_list !! n
a078495_list = [1, 1, 1, 1, 1, 1, 1] ++
  zipWith div (foldr1 (zipWith (+)) (map b [1,3])) a078495_list
  where b i = zipWith (*) (drop i a078495_list) (drop (7-i) a078495_list)
-- Reinhard Zumkeller, May 05 2013

I:=[1,1,1,1,1,1,1]; [n le 7 select I[n] else (Self(n-1)*Self(n-6) + Self(n-3)*Self(n-4))/Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 21 2018

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==1,a[n] == (a[n-1]*a[n-6]+a[n-3]*a[n-4])/a[n-7]},a,{n,40}] (* Harvey P. Dale, Apr 20 2012 *)

{a(n) = if( n<0, a(6-n), if( n<7, 1, (a(n-1) * a(n-6) + a(n-3) * a(n-4)) / a(n-7)))};

{a(n) = 2^(b(n-9) + b(n-7)) * 3^b(n-8)}; {b(n) = (n^2 + 10*n + 1 - n%2*13) \ 60 + 1}; /* b(n) = A025795(n) */

a(n) = 144 * a(n-6) * a(n-10) / a(n-16), a(n) = a(6-n) for all n in Z.

A335106 Irregular triangle T(n,k) is the number of times that prime(k) is the greatest part in a partition of n into prime parts; Triangle T(n,k), n>=0, 1 <= k <= max(1,A000720(A335285(n))), read by rows.

Original entry on oeis.org

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

Offset: 0

David James Sycamore, Michael De Vlieger, May 23 2020

Let L(n) be the set of least part primes of all prime partitions of n, and let G(n) be corresponding set of greatest prime parts. All prime partitions, commencing with p in L(n) and terminating with q in G(n) can be shown as unique pathways on a partition tree of n; see link for details. |G(n)| = A000607(n).

			A000607(10) = 5 and the prime partitions of 10 are: (2,2,2,2,2), (2,2,3,3), (2,3,5), (5,5) and (3,7). Thus G(10) = {2,3,5,5,7}, and consequently row 10 is [1,1,2,1]. In the table below, for n >= 2,  0 is used to indicate when prime(k) is not in G(n) and is less than the greatest member of G(n), otherwise the entry for prime(k) not in G(n) is left empty. For n >= 2 the sum of entries in the n-th row is |G(n)| = A000607(n). Triangle T(n,k) begins:
0;
0;
1;
0, 1;
1;
0, 1, 1;
1, 1;
0, 1, 1, 1;
1, 1, 1;
0, 2, 1, 1;
1, 1, 2, 1;
0, 2, 2, 1, 1;
1, 2, 2, 2;
0, 2, 3, 2, 1, 1;
1, 2, 3, 3, 1;
0, 3, 4, 3, 1, 1;
1, 2, 4, 4, 2, 1;
0, 3, 5, 5, 2, 1, 1;
...

David James Sycamore, Prime Partition Trees

Cf. A000040, A000607, A000720, A333365, A331634, A335285.
Row sums gives A000607 for n > 1. Length of n-th row is A000720(A335285(n)) for n >1.
Number of partition of n in the first k primes: A059841 (k = 1), A103221 (k = 2), A025795 (k = 3), A029144 (k = 4), A140952 (k = 5), A140953 (k = 6).

Flatten@ Block[{nn = 22, t}, t = Block[{s = {Prime@ PrimePi@ nn}}, KeySort@ Merge[#, Identity] &@ Join[{0 -> {}, 1 -> {}}, Reap[Do[If[# <= nn, Sow[# -> s]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1] ] ] &@Total[s], {i, Infinity}]][[-1, -1]] ] ]; Array[Function[p, If[! IntegerQ@ First@ p, {0}, Array[Count[p, Prime@ #] &, PrimePi@ Max@ p]]]@ Map[Max, t[[#]]] &, Max@ Keys@ t]] (* Michael De Vlieger, May 23 2020 *)
row[0]={0}; row[k_] := Join[If[OddQ@k, {0}, {}], Last /@ Tally@ Sort[ First /@ IntegerPartitions[k, All, Prime@ Range@ PrimePi@ k]]]; Join @@ Array[row, 20, 0] (* Giovanni Resta, May 31 2020 *)

More terms from Giovanni Resta, May 31 2020

A140952 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 25, 28, 31, 35, 39, 43, 48, 53, 58, 64, 70, 77, 84, 91, 100, 108, 117, 127, 137, 148, 159, 172, 184, 198, 212, 227, 243, 259, 277, 295, 314, 334, 355, 377, 400, 424, 449, 475, 502, 531, 560

Offset: 0

Alois P. Heinz, Jul 25 2008

Number of partitions of n into the first 5 primes.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,0,0,-1,-1,0,1,0,-1,0,1,0,-1,0,1,1,0,0,0,0,-1,-1,0,1).

Cf. A000040, A025795, A029144, A103221, A140953, A335106.

M := Matrix(28, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 3, 11, 15, 19, 20, 28])) then 1 elif j=1 and member(i, [8, 9, 13, 17, 25, 26]) then -1 else 0 fi):
a:= n-> (M^(n))[1,1]:
seq(a(n), n=0..50);

CoefficientList[Series[1/Times@@(1-x^Prime[Range[5]]),{x,0,70}],x] (* or *) LinearRecurrence[{0,1,1,0,0,0,0,-1,-1,0,1,0,-1,0,1,0,-1,0,1,1,0,0,0,0,-1,-1,0,1},{1,0,1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,15,17,19,22,25,28,31,35,39,43,48},70] (* Harvey P. Dale, Jun 18 2021 *)

A140953 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)*(1-x^13)).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 16, 19, 21, 25, 28, 32, 36, 41, 46, 52, 58, 65, 72, 80, 89, 98, 109, 119, 132, 144, 158, 173, 189, 206, 224, 244, 264, 287, 310, 336, 362, 391, 421, 453, 487, 523, 561, 601, 644, 688, 736, 785, 838, 893

Offset: 0

Alois P. Heinz, Jul 25 2008

Number of partitions of n into the first 6 primes. [Corrected by Harvey P. Dale, Dec 05 2022]

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0, 0,0,0,-1,-1,0,1,0,0,0,0,-1,-1,0,1,1,1,1,0,-1,-1,0,0,0,0,1,0,-1,-1,0,0,0,0,1,1,0,-1).

Cf. A000040, A025795, A029144, A103221, A140952, A335106.

M := Matrix(41, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 3, 11, 19, 20, 21, 22, 30, 38, 39])) then 1 elif j=1 and member(i, [8, 9, 16, 17, 24, 25, 32, 33, 41]) then -1 else 0 fi):
a:= n -> (M^(n))[1,1]:
seq(a(n), n=0..50);

CoefficientList[Series[1/Times@@Table[1-x^p,{p,Prime[Range[6]]}],{x,0,60}],x] (* or *) LinearRecurrence[{0,1,1,0,0,0,0,-1,-1,0,1,0,0,0,0,-1,-1,0,1,1,1,1,0,-1,-1,0,0,0,0,1,0,-1,-1,0,0,0,0,1,1,0,-1},{1,0,1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,16,19,21,25,28,32,36,41,46,52,58,65,72,80,89,98,109,119,132,144,158,173,189,206},70] (* Harvey P. Dale, Dec 05 2022 *)

A115296 Skew version of correlation triangle for constant sequence 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 2, 3, 2, 1, 0, 0, 0, 0, 1, 3, 3, 2, 1, 0, 0, 0, 0, 0, 2, 4, 3, 2, 1, 0, 0, 0, 0, 0, 1, 3, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 2, 4, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 4, 5, 4, 3, 2, 1

Offset: 0

Paul Barry, Jan 19 2006

Row sums are A001399. Diagonal sums are A025795.

			Triangle begins
1;
0,1;
0,1,1;
0,0,2,1;
0,0,1,2,1;
0,0,0,2,2,1;
0,0,0,1,3,2,1;
0,0,0,0,2,3,2,1;

G.f.: 1/((1-x*y)*(1-x^2*y)*(1-x^3*y^2)); Number triangle T(n, k)=sum{j=0..k, [j<=n-k]*[j<=2k-n]}; T(n, k)=A003983(k, n-k).

cp's OEIS Frontend

A008667 Expansion of g.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A078495 a(n) = (a(n-1) * a(n-6) + a(n-3) * a(n-4)) / a(n-7) (a variant of Somos-7).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

Formula

A335106 Irregular triangle T(n,k) is the number of times that prime(k) is the greatest part in a partition of n into prime parts; Triangle T(n,k), n>=0, 1 <= k <= max(1,A000720(A335285(n))), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A140952 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^11)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A140953 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^11)*(1-x^13)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A115296 Skew version of correlation triangle for constant sequence 1.

Views

Author

Keywords

Comments

Examples

Formula

A008667 Expansion of g.f.: 1/((1-x^2)(1-x^3)(1-x^4)*(1-x^5)).

A140952 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)).

A140953 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)*(1-x^13)).