A001402 -id:A001402 - cp's OEIS frontend

A160647 Self-convolution of sequence A001402.

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

A026820 Euler's table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <= k for 1 <= k <= n. Also number of partitions of n into at most k parts.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 11, 13, 14, 15, 1, 5, 10, 15, 18, 20, 21, 22, 1, 5, 12, 18, 23, 26, 28, 29, 30, 1, 6, 14, 23, 30, 35, 38, 40, 41, 42, 1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56, 1, 7, 19, 34, 47, 58, 65, 70, 73, 75, 76, 77

Offset: 1

Clark Kimberling

			Triangle starts:
  1;
  1, 2;
  1, 2,  3;
  1, 3,  4,  5;
  1, 3,  5,  6,  7;
  1, 4,  7,  9, 10, 11;
  1, 4,  8, 11, 13, 14, 15;
  1, 5, 10, 15, 18, 20, 21, 22;
  1, 5, 12, 18, 23, 26, 28, 29, 30;
  1, 6, 14, 23, 30, 35, 38, 40, 41, 42;
  1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56;
  ...

G. Chrystal, Algebra, Vol. II, p. 558.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.

Alois P. Heinz, Robert G. Wilson v, Rows n = 1..141, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 831. [scanned copy]
Carolyn Echter, Georg Maier, Juan-Diego Urbina, Caio Lewenkopf, and Klaus Richter, Many-body density of states of bosonic and fermionic gases: a combinatorial approach, arXiv:2409.08696 [cond-mat.quant-gas], 2024. See p. 10.
L. Euler, Introductio in Analysin Infinitorum, Book I, chapter XVI.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers.
R. Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, Univ. Wien, 2013.
Sergei Viznyuk, C program.
Sergei Viznyuk, Local copy of C program.
Eric Weisstein's World of Mathematics, Partition Function q.
Index entries for sequences related to partitions - _Reinhard Zumkeller_, Jan 21 2010

Partial sums of rows of A008284, row sums give A058397, central terms give A171985, mirror is A058400.
T(n,n) = A000041(n), T(n,1) = A000012(n), T(n,2) = A008619(n) for n>1, T(n,3) = A001399(n) for n>2, T(n,4) = A001400(n) for n>3, T(n,5) = A001401(n) for n>4, T(n,6) = A001402(n) for n>5, T(n,7) = A008636(n) for n>6, T(n,8) = A008637(n) for n>7, T(n,9) = A008638(n) for n>8, T(n,10) = A008639(n) for n>9, T(n,11) = A008640(n) for n>10, T(n,12) = A008641(n) for n>11, T(n,n-2) = A007042(n-1) for n>2, T(n,n-1) = A000065(n) for n>1.
Cf. A008284, A026840, A134737, A173519.

import Data.List (inits)
a026820 n k = a026820_tabl !! (n-1) !! (k-1)
a026820_row n = a026820_tabl !! (n-1)
a026820_tabl = zipWith
   (\x -> map (p x) . tail . inits) [1..] $ tail $ inits [1..] where
   p 0 _ = 1
   p _ [] = 0
   p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
-- Reinhard Zumkeller, Dec 18 2013

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Apr 21 2012

t[n_, k_] := Length@ IntegerPartitions[n, k]; Table[ t[n, k], {n, 12}, {k, n}] // Flatten
(* Second program: *)
T[n_, k_] := T[n, k] = If[n==0 || k==1, 1, T[n, k-1] + If[k>n, 0, T[n-k, k]]]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)

T(n,k)=my(s); forpart(v=n,s++,,k); s \\ Charles R Greathouse IV, Feb 27 2018

from sage.combinat.partition import number_of_partitions_length
from itertools import accumulate
for n in (1..11):
    print(list(accumulate([number_of_partitions_length(n, k) for k in (1..n)])))
# Peter Luschny, Jul 28 2022

T(T(n,n),n) = A134737(n). - Reinhard Zumkeller, Nov 07 2007
T(A000217(n),n) = A173519(n). - Reinhard Zumkeller, Feb 20 2010
T(n,k) = T(n,k-1) + T(n-k,k). - Thomas Dybdahl Ahle, Jun 13 2011
T(n,k) = Sum_{i=1..min(k,floor(n/2))} T(n-i,i) + Sum_{j=1+floor(n/2)..k} A000041(n-j). - Bob Selcoe, Aug 22 2014 [corrected by Álvar Ibeas, Mar 15 2018]
O.g.f.: Product_{i>=0} 1/(1-y*x^i). - Geoffrey Critzer, Mar 11 2012
T(n,k) = A008284(n+k,k). - Álvar Ibeas, Jan 06 2015

A001592 Hexanacci numbers: a(n+1) = a(n)+...+a(n-5) with a(0)=...=a(4)=0, a(5)=1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, 463968, 920319, 1825529, 3621088, 7182728, 14247536, 28261168, 56058368, 111196417, 220567305, 437513522, 867844316, 1721441096, 3414621024

Offset: 0

N. J. A. Sloane

a(n+5) is the number of ways of throwing n with an unstated number of standard dice and so the row sum of A061676; for example a(9)=8 is the number of ways of throwing a total of 4: 4, 3+1, 2+2, 1+3, 2+1+1, 1+2+1, 1+1+2 and 1+1+1+1; if order did not distinguish partitions (i.e. the dice were indistinguishable) then this would produce A001402 instead. - Henry Bottomley, Apr 01 2002
Number of permutations (p(i)) [of the numbers 1 to n, presumably? - N. J. A. Sloane, Jan 22 2021] satisfying -k<=p(i)-i<=r, i=1..n-5, with k=1, r=5. - Vladimir Baltic, Jan 17 2005
a(n+5) is the number of compositions of n with no part greater than 6. - Vladimir Baltic, Jan 17 2005
Equivalently, for n>=0: a(n+6) is the number of binary strings with length n where at most 5 ones are consecutive, see fxtbook link below. - Joerg Arndt, Apr 08 2011

Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
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).

Indranil Ghosh, Table of n, a(n) for n = 0..3361 (terms 0..200 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), pp. 307-309
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 13
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Hexanacci Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1).

Row 6 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).

CoefficientList[Series[x^5/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6), {x, 0, 50}], x]
a[0] = a[1] = a[2] = a[3] = a[4] = 0; a[5] = a[6] = 1; a[n_] := a[n] = 2 a[n - 1] - a[n - 7]; Array[a, 36]
LinearRecurrence[{1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,1,1,1,1,1]^n*[0;0;0;0;0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

a(n)= my(x='x, p=polrecip(1 - x - x^2 - x^3 - x^4 - x^5 - x^6)); polcoef(lift(Mod(x,p)^n),5);
vector(31,n,a(n-1)) \\ Joerg Arndt, May 16 2021

G.f.: x^5/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6). - Simon Plouffe in his 1992 dissertation
G.f.: Sum_{n >= 0} x^(n+5) * [ Product_{k = 1..n} (k + k*x + k*x^2 + k*x^3 + k*x^4 + x^5)/(1 + k*x + k*x^2 + k*x^3 + k*x^4 + k*x^5) ]. - Peter Bala, Jan 04 2015
Another form of the g.f.: f(z) = (z^5-z^6)/(1-2*z+z^7); then a(n) = Sum_((-1)^i*binomial(n-5-6*i,i)*2^(n-5-7*i), i=0..floor((n-5)/7))-Sum_((-1)^i*binomial(n-6-6*i,i)*2^(n-6-7*i), i=0..floor((n-6)/7)) with Sum_(alpha(i), i=m..n) = 0 for m>n. - Richard Choulet, Feb 22 2010
Sum_{k=0..5*n} a(k+b)*A063260(n,k) = a(6*n+b), b>=0.
a(n) = 2*a(n-1)-a(n-7). - Vincenzo Librandi, Dec 19 2010
lim n-> oo a(n)/a(n-1) = A118427. - R. J. Mathar, Mar 11 2024

More terms from Robert G. Wilson v, Nov 16 2000

A026812 Number of partitions of n in which the greatest part is 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 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, 5427, 5942, 6510, 7104, 7760

Offset: 0

Clark Kimberling

Also number of partitions of n into 6 parts. - Washington Bomfim, Jan 15 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
G. E. Andrews, Partitions: At the Interface of q-Series and Modular Forms, The Ramanujan Journal 7, 385-400 (2003), Eq.(3.10).
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).

Essentially same as A001402.
Cf. A026810, A026811, A026813, A026814, A026815, A026816.
Cf. A001401, A001402.

List([0..70],n->NrPartitions(n,6)); # Muniru A Asiru, May 17 2018

Table[ Length[ Select[ Partitions[n], First[ # ] == 6 & ]], {n, 1, 60} ]
CoefficientList[Series[x^6/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Drop[LinearRecurrence[{1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1}, Append[Table[0,{20}],1],115],14] (* Robert A. Russell, May 17 2018 *)

my(x='x+O('x^99)); concat(vector(6), Vec(x^6/prod(k=1, 6, 1-x^k))) \\ Altug Alkan, May 17 2018

a = vector(60,n,n--; round((n+11)*((6*n^4+249*n^3+2071*n^2 -4931*n+40621) /518400 +n\2*(n+10)/192+((n+1)\3+n\3*2)/54))); a = concat([0,0,0,0,0,0], a) \\ Washington Bomfim, Jan 16 2021

G.f.: x^6 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)). - Colin Barker, Dec 20 2012
a(n) = A008284(n,6). - Robert A. Russell, May 13 2018
a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} 1. - Wesley Ivan Hurt, Jun 29 2019
a(n) = A001402(n) - A001401(n). a(n) = A001402(n-6). - Washington Bomfim, Jan 15 2021
a(n) = round((1/86400)*n^5 + (1/3840)*n^4 + (19/12960)*n^3 - (n mod 2)*(1/384)*n^2 + (1/17280)*b(n mod 6)*n), where b(0)=96, b(1)=b(5)=-629, b(2)=b(4)=-224, and b(3)=-309. - Washington Bomfim and Jon E. Schoenfield, Jan 16 2021

More terms from Robert G. Wilson v, Jan 11 2002

a(0)=0 prepended by Seiichi Manyama, Jun 08 2017

A037145 Expansion of 1/((1-x^2)(1-x^3)...(1-x^6)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 6, 6, 9, 9, 14, 13, 19, 20, 26, 27, 36, 36, 47, 49, 60, 63, 78, 80, 97, 102, 120, 126, 149, 154, 180, 189, 216, 227, 260, 270, 307, 322, 361, 378, 424, 441, 492, 515, 568, 594, 656, 682, 750

Offset: 0

N. J. A. Sloane

Also, Molien series for invariants of finite Coxeter group A_5. The Molien series for the finite Coxeter group of type A_k (k >= 1) has G.f. = 1/Prod_{i=2..k+1} (1-x^i). - N. J. A. Sloane, Jan 11 2016

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -2, -2, -1, 0, 1, 2, 2, 0, 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. A001402 (partial sums).

CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,6}],{x,0,50}],x] (* Harvey P. Dale, Dec 25 2012 *)

A288341 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^6)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 44, 64, 90, 125, 169, 227, 298, 388, 498, 634, 797, 996, 1231, 1513, 1844, 2235, 2689, 3221, 3833, 4542, 5353, 6284, 7341, 8547, 9907, 11447, 13176, 15121, 17293, 19725, 22427, 25436, 28767, 32459, 36529, 41023, 45958, 51385, 57327

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 6 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -2, 2, 1, 0, 0, 0, -1, -2, 2, -1, 1, 0, 1, 0, -2, 1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), this sequence (k=6), A288342 (k=7), A288343 (k=8), A288344 (k=9), A288345 (k=10).

Cf. A288253. Column 6 of A092905. A001402 (first differences).

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 6, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A008636 Number of partitions of n into at most 7 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 49, 65, 82, 105, 131, 164, 201, 248, 300, 364, 436, 522, 618, 733, 860, 1009, 1175, 1367, 1579, 1824, 2093, 2400, 2738, 3120, 3539, 4011, 4526, 5102, 5731, 6430, 7190, 8033, 8946, 9953, 11044, 12241, 13534, 14950, 16475, 18138

Offset: 0

N. J. A. Sloane, Mar 15 1996

Also, the number of partitions of n into parts <= 7: a(n) = A026820(n, 7). - Reinhard Zumkeller, Jan 21 2010
Counts unordered closed walks of weight n on a single vertex graph with 7 loops of weights 1, 2, 3, 4, 5, 6 and 7. - David Neil McGrath, Apr 11 2015
Number of different distributions of n+28 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - Ece Uslu and Esin Becenen, Jan 11 2016

			There are 28 partitions of 9 into parts less than or equal to 7. These are (72)(711)(63)(621)(6111)(54)(531)(522)(5211)(51111)(441)(432)(4311)(4221)(42111)(411111)(333)(3321)(33111)(3222)(32211)(321111)(3111111)(22221)(222111)(2211111)(21111111)(111111111). - _David Neil McGrath_, Apr 11 2015
a(3) = 3, i.e., {1,2,3,4,5,7,9}, {1,2,3,4,6,7,8}, {1,2,3,4,5,6,10}. Number of different distributions of 31 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - _Ece Uslu_, Esin Becenen, Jan 11 2016

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 356
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. A008284, A026820.

with(combstruct):ZL8:=[S,{S=Set(Cycle(Z,card<8))}, unlabeled]: seq(count(ZL8,size=n),n=0..48); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=7)},unlabelled]: seq(combstruct[count](B, size=n), n=0..48); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 7} ], {x, 0, 60} ], x ]

{a(n)=(2*n^6+168*n^5+5530*n^4+90160*n^3+754299*n^2+(2988020+44800*(1-n%3))*n+6654375+1575*(3*n^2+84*n+511)*(-1)^n)\7257600}; \\ Tani Akinari, May 27 2014

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).
a(n) = A008284(n+7, 7), n >= 0.
a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-8) + a(n-10) + a(n-11) + 2*a(n-12) - 2*a(n-16) - a(n-17) - a(n-18) + a(n-20) + a(n-21) + a(n-23) - a(n-26) - a(n-27) + a(n-28). - David Neil McGrath, Apr 11 2015
a(n+7) = a(n) + A001402(n). - Ece Uslu, Esin Becenen, Jan 11 2016
a(n) = A026813(n+7). - R. J. Mathar, Feb 13 2019
From Vladimír Modrák, Jul 30 2022: (Start)
a(n) = Sum_{p=0..floor(n/7)} Sum_{m=0..floor(n/6)} Sum_{k=0..floor(n/5)} Sum_{j=0..floor(n/4)} Sum_{i=0..floor(n/3)} ceiling((max(0, n + 1 - 3*i - 4*j - 5*k - 6*m - 7*p))/2).
a(n) = Sum_{m=0..floor(n/7)} Sum_{k=0..floor(n/6)} Sum_{j=0..floor(n/5)} Sum_{i=0..floor(n/4)} floor(((max(0, n + 3 - 4*i - 5*j - 6*k - 7*m))^2+4)/12). (End)

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

A371070 a(n) is the number of distinct volumes > 0 of tetrahedra with the sum of their integer edge lengths equal to n.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 2, 3, 6, 5, 7, 12, 10, 16, 19, 21, 26, 34, 37, 44, 56, 60, 67, 93, 92, 111, 137, 140, 166, 192, 211, 246, 279, 306, 333, 392, 428, 464, 538, 565, 627, 709, 768, 826, 939, 998, 1089, 1230, 1312, 1403, 1590, 1658, 1798, 1987, 2088, 2266, 2495

Offset: 6

Hugo Pfoertner, Mar 18 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 6..200
Hugo Pfoertner, Plot of ratio a(n)/A208454(n), using Plot 2. Is the asymptotic ratio for n->oo finite or 0?

Cf. A001402, A097125, A208454, A346575, A371071, A371345.

a371070(n) = {my (L=List()); forpart (w=n, forperm (w,v, if(v[4]+v[5]0, listput (L,CM))), [1,n], [6,6]); #Set(Vec(L))};

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_permutations
def A371070(n):
    CM = lambda x,y,z,t,u,v: (x*y*z<<2)+(a:=x+y-t)*(b:=x+z-u)*(c:=y+z-v)-x*c**2-y*b**2-z*a**2
    TR1 = lambda x,y,z: not(x+y0 and M not in d:
                    d.add(M)
                    c += 1
    return c # Chai Wah Wu, Mar 23 2024

a(n) <= A208454(n).

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

A256226 Number of partitions of 6n into 6 parts.

Original entry on oeis.org

0, 1, 11, 58, 199, 532, 1206, 2432, 4494, 7760, 12692, 19858, 29941, 43752, 62239, 86499, 117788, 157532, 207338, 269005, 344534, 436140, 546261, 677571, 832989, 1015691, 1229120, 1476997, 1763332, 2092435, 2468926, 2897747, 3384171, 3933815, 4552649, 5247008

Offset: 0

Colin Barker, Mar 19 2015

			For n=2, the 11 partitions of 12 are Xs = [7,1,1,1,1,1], [6,2,1,1,1,1], [5,3,1,1,1,1], [4,4,1,1,1,1], [5,2,2,1,1,1], [4,3,2,1,1,1], [3,3,3,1,1,1], [4,2,2,2,1,1], [3,3,2,2,1,1], [3,2,2,2,2,1] and [2,2,2,2,2,2].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-3,-3,5,0,-5,4,-1).

Cf. A001402, A077043, A238340, A256225.

CoefficientList[Series[x (3 x^7 + 14 x^6 + 21 x^5 + 21 x^4 + 22 x^3 + 19 x^2 + 7 x + 1) / ((x - 1)^6 (x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)

concat(0, Vec(x*(3*x^7+14*x^6+21*x^5+21*x^4+22*x^3+19*x^2+7*x+1)/((x-1)^6*(x+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))

concat(0, vector(35, n, k=0; forpart(p=6*n, k++, , [6,6]); k)) \\ Colin Barker, Mar 21 2015

G.f.: x*(3*x^7+14*x^6+21*x^5+21*x^4+22*x^3+19*x^2+7*x+1) / ((x-1)^6*(x+1)*(x^4+x^3+x^2+x+1)).

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