A097701 -id:A097701 - cp's OEIS frontend

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

0, 0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166, 192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762, 819, 889, 952, 1029, 1099, 1183, 1260, 1352, 1436, 1536, 1628, 1736, 1836, 1953, 2061, 2187, 2304, 2439

Offset: 0

N. J. A. Sloane, Simon Plouffe, Stephen P. Humphries

Number of 2 X 2 square partitions of n.
1/((1-x^2)*(1-x^4)^2*(1-x^6)) is the Molien series for 4-dimensional representation of a certain group of order 192 [Nebe, Rains, Sloane, Chap. 7].
Number of ways of writing n as n = p+q+r+s so that p >= q, p >= r, q >= s, r >= s with p, q, r, s >= 1. That is, we can partition n as
pq
rs
with p >= q, p >= r, q >= s, r >= s.
The coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) is a(n+4), where s(n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding to the two row partition and * represents the inner or Kronecker product of symmetric functions. - Mike Zabrocki, Dec 22 2005
Let F() be the Fibonacci sequence A000045. Let f([x, y, z, w]) = F(x) * F(y) * F(z) * F(w). Let N([x, y, z, w]) = x^2 + y^2 + z^2 + w^2. Let Q(k) = set of all ordered quadruples of integers [x, y, z, w] such that 1 <= x <= y <= z <= w and N([x, y, z, w]) = k. Let P(n) = set of all unordered triples {q1, q2, q3} of elements of some Q(k) such that max(w1, w2, w3) = n and f(q1) + f(q2) = f(q3). Then a(n-1) is the number of elements of P(n). - Michael Somos, Jan 21 2015
Number of partitions of 2n+2 into 4 parts with alternating parity from smallest to largest (or vice versa). - Wesley Ivan Hurt, Jan 19 2021

			a(7) = 4:
41 32 31 22
11 11 21 21
G.f. = x^4 + x^5 + 3*x^6 + 4*x^7 + 7*x^8 + 9*x^9 + 14*x^10 + 17*x^11 + ...
a(5-1) = 1 because P(5) has only one triple {[1,1,1,5], [2,2,2,4], [1,3,3,3]} of elements from Q(28) where f([1,1,1,5]) = 5, f([2,2,2,4]) = 3, f([1,3,3,3]) = 8, and 5 + 3 = 8. - _Michael Somos_, Jan 21 2015
a(6-1) = 1 because P(6) has only one triple {[1,1,2,6], [2,2,3,5], [1,3,4,4]} of elements from Q(42) where f([1,1,2,6]) = 8, f([2,2,3,5]) = 10, f([1,3,4,4]) = 18 and 8 + 10 = 18. - _Michael Somos_, Jan 21 2015
a(7-1) = 3 because P(7) has three triples. The triple {[1,1,1,7], [2,4,4,4], [3,3,3,5]} from Q(52) where f([1,1,1,7]) = 13, f([2,4,4,4]) = 27, f([3,3,3,5]) = 40 and 13 + 27 = 40. The triple {[1,2,2,7], [2,3,3,6], [1,4,4,5]} from Q(58) where f([1,2,2,7]) = 13, f([2,3,3,6]) = 32, f([1,4,4,5]) = 45 and 13 + 32 = 45. The triple {[1,1,3,7], [2,2,4,6], [1,3,5,5]} from Q(60) where f([1,1,3,7]) = 26, f([2,2,4,6]) = 24, f([1,3,5,5]) = 50 and 26 + 24 = 50. - _Michael Somos_, Jan 21 2015

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis VIII: Plane partition diamonds, Advances Applied Math., 27 (2001), 231-242 (Cor. 2.1, n=1).
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

T. D. Noe, Table of n, a(n) for n = 0..1000
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 5.
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2. [MR1219862 (94d:11029)]
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
S. P. Humphries, Home page
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 232
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Michael Somos, In the Elliptic Realm
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

See A266769 for a version without the four leading zeros.
Cf. A001993, A070557, A070558, A070559, A089299, A001970, A089292, A026810, A001400.
First differences of A097701.
Cf. A082424, A082437.

a:=[0,0,0,0,1,1,3,4];; for n in [9..60] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]-2*a[n-4]-a[n-5]+2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Sep 10 2019

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(H);

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( x^4/((1-x)*(1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 10 2019

a:= n-> (Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [1,2,-1,-2,-1,2,1,-1][i] else 0 fi)^n)[1,5]: seq(a(n), n=0..60); # Alois P. Heinz, Jul 31 2008

CoefficientList[Series[x^4/((1-x)*(1-x^2)^2*(1-x^3)), {x,0,60}], x] (* Jean-François Alcover, Mar 30 2011 *)
LinearRecurrence[{1,2,-1,-2,-1,2,1,-1},{0,0,0,0,1,1,3,4},60] (* Harvey P. Dale, Mar 04 2012 *)
a[ n_]:= Quotient[9(n+1)(-1)^n +2n^3 -9n +65, 144]; (* Michael Somos, Jan 21 2015 *)
a[ n_]:= Sign[n] SeriesCoefficient[ x^4/((1-x)(1-x^2)^2(1-x^3)), {x, 0, Abs@n}]; (* Michael Somos, Jan 21 2015 *)

{a(n) = (9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65) \ 144}; /* Michael Somos, Jan 21 2015 */

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,1,2,-1,-2,-1,2,1]^n*[0;0;0;0;1;1;3;4])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

def AA008763_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^4/((1-x)*(1-x^2)^2*(1-x^3))).list()
AA008763_list(60) # G. C. Greubel, Sep 10 2019

Let f4(n) = number of partitions n = p+q+r+s into exactly 4 parts, with p >= q >= r >= s >= 1 (see A026810, A001400) and let g4(n) be the number with q > r (so that g4(n) = f4(n-2)). Then a(n) = f4(n) + g4(n).
a(n) = (1/144)*( 2*n^3 + 9*n*((-1)^n - 1) - 16*((n is 2 mod 3) - (n is 1 mod 3)) ).
a(n) = (1/72)*(n+3)*(n+2)*(n+1)-(1/12)*(n+2)*(n+1)+(5/144)*(n+1)+(1/16)*(n+1)*(-1)^n+(1/16)*(-1)^(n+1)+(7/144)+(2*sqrt(3)/27)*sin(2*Pi*n/3). - Richard Choulet, Nov 27 2008
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8), n>7. - Harvey P. Dale, Mar 04 2012
a(n) = floor((9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65)/144). - Tani Akinari, Nov 06 2012
a(n+1) - a(n) = A008731(n-3). - R. J. Mathar, Aug 06 2013
a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 21 2015
Euler transform of length 3 sequence [1, 2, 1]. - Michael Somos, Jun 26 2017

Entry revised Dec 25 2003

A115262 Correlation triangle for n+1.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 8, 8, 4, 5, 11, 14, 11, 5, 6, 14, 20, 20, 14, 6, 7, 17, 26, 30, 26, 17, 7, 8, 20, 32, 40, 40, 32, 20, 8, 9, 23, 38, 50, 55, 50, 38, 23, 9, 10, 26, 44, 60, 70, 70, 60, 44, 26, 10, 11, 29, 50, 70, 85, 91, 85, 70, 50, 29, 11

Offset: 0

Paul Barry, Jan 18 2006

This sequence (formatted as a square array) gives the counts of all possible squares in an m X n rectangle. For example, 11 = 8 (1 X 1 squares) + 3 (2 X 2 square) in 4 X 2 rectangle. - Philippe Deléham, Nov 26 2009
From Clark Kimberling, Feb 07 2011: (Start)
Also the accumulation array of min{n,k}, when formatted as a rectangle.
This is the accumulation array of the array M=A003783 given by M(n,k)=min{n,k}; see A144112 for the definition of accumulation array.
The accumulation array of A115262 is A185957. (End)
From Clark Kimberling, Dec 22 2011: (Start)
As a square matrix, A115262 is the self-fusion matrix of A000027 (1,2,3,4,...).  See A193722 for the definition of fusion and A202673 for characteristic polynomials associated with A115622. (End)

			Triangle begins
  1;
  2,  2;
  3,  5,  3;
  4,  8,  8,  4;
  5, 11, 14, 11,  5;
  6, 14, 20, 20, 14,  6;
  ...
When formatted as a square matrix:
  1,  2,  3,  4,  5, ...
  2,  5,  8, 11, 14, ...
  3,  8, 14, 20, 26, ...
  4, 11, 20, 30, 40, ...
  5, 14, 26, 40, 55, ...
  ...

Cf. A000027, A202673, A271916.
For the triangular version: row sums are A001752. Diagonal sums are A097701. T(2n,n) is A000330(n+1).
Diagonals (1,5,...): A000330 (square pyramidal numbers),
diagonals (2,8,...): A007290,
diagonals (3,11,...): A051925,
diagonals (4,14,...): A159920,
antidiagonal sums: A001752.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
(* Clark Kimberling, Dec 22 2011 *)

Let f(m,n) = m*(m-1)*(3*n-m-1)/6. This array is (with a different offset) the infinite square array read by antidiagonals U(m,n) = f(n,m) if m < n, U(m,n) = f(m,n) if m <= n. See A271916. - N. J. A. Sloane, Apr 26 2016
G.f.: 1/((1-x)^2*(1-x*y)^2*(1-x^2*y)).
Number triangle T(n, k) = Sum_{j=0..n} [j<=k]*(k-j+1)[j<=n-k]*(n-k-j+1).
T(2n,n) - T(2n,n+1) = n+1.

A002625 Expansion of 1/((1-x)^3(1-x^2)^2(1-x^3)).

Original entry on oeis.org

1, 3, 8, 17, 33, 58, 97, 153, 233, 342, 489, 681, 930, 1245, 1641, 2130, 2730, 3456, 4330, 5370, 6602, 8048, 9738, 11698, 13963, 16563, 19538, 22923, 26763, 31098, 35979, 41451, 47571, 54390, 61971, 70371, 79660, 89901, 101171, 113540, 127092, 141904, 158068, 175668, 194804, 215568

Offset: 0

N. J. A. Sloane

Number of (integer) partitions of n into 3 sorts of 1's, 2 sorts of 2's, and 1 sort of 3's. - Joerg Arndt, May 17 2013

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 205
Gerzson Keri and Patric R. J. Östergård, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.
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
Index entries for linear recurrences with constant coefficients, signature (3,-1,-4,2,2,2,-4,-1,3,-1).

Partial sums of A097701.

A002625:=1/(z**2+z+1)/(z+1)**2/(z-1)**6; [Simon Plouffe in his 1992 dissertation.]

CoefficientList[Series[1/((1-x)^3*(1-x^2)^2*(1-x^3)),{x,0,50}],x] (* Vincenzo Librandi, Feb 25 2012 *)
LinearRecurrence[{3,-1,-4,2,2,2,-4,-1,3,-1},{1,3,8,17,33,58,97,153,233,342},50] (* Harvey P. Dale, Sep 24 2022 *)

Vec(1/(1-x)^3/(1-x^2)^2/(1-x^3)+O(x^99)) \\ Charles R Greathouse IV, Apr 30 2012

a(n) = floor((n+1)*(135*(-1)^n + 6*n^4 + 144*n^3 + 1256*n^2 + 4744*n + 6785)/8640+1/2). - Tani Akinari, Oct 07 2012

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

A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 9, 1, 1, 12, 2, 1, 16, 5, 1, 20, 9, 1, 25, 16, 1, 30, 25, 1, 36, 39, 1, 1, 42, 56, 2, 1, 49, 80, 5, 1, 56, 109, 10, 1, 64, 147, 19, 1, 72, 192, 32, 1, 81, 249, 54, 1, 90, 315, 84, 1, 100, 396, 129, 1, 1, 110, 489, 190, 2, 1, 121, 600, 275, 5

Offset: 1

Gus Wiseman, May 21 2022

			Triangle begins:
   1
   1   1
   1   2
   1   4
   1   6
   1   9   1
   1  12   2
   1  16   5
   1  20   9
   1  25  16
   1  30  25
   1  36  39   1
   1  42  56   2
   1  49  80   5
   1  56 109  10
For example, row n = 7 counts the following partitions:
  (1111111)  (7)       (43)
             (52)      (331)
             (61)
             (322)
             (421)
             (511)
             (2221)
             (3211)
             (4111)
             (22111)
             (31111)
             (211111)

Row sums are A000041.
Row lengths are A000194, reversed A003056.
Column k = 1 is A002620, reversed A238875.
Column k = 2 is A097701.
The version for permutations is A008292, opposite A123125.
The weak version is A115720/A115994, rank statistic A257990.
The version for compositions is A352524, weak A352525.
The version for reversed partitions is A353319.
A000700 counts self-conjugate partitions, ranked by A088902.
A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
A238352 counts reversed partitions by fixed points, rank statistic A352822.
Cf. A000701, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.

partsabove[y_]:=Length[Select[Range[Length[y]],#

A164678 Convolve A008619 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

Original entry on oeis.org

1, 2, 1, 5, 4, 2, 9, 8, 6, 2, 17, 16, 14, 9, 3, 28, 27, 25, 20, 12, 3, 47, 46, 44, 39, 30, 16, 4, 73, 72, 70, 65, 56, 40, 20, 4, 114, 113, 111, 106, 97, 80, 55, 25, 5, 170, 169, 167, 162, 153, 136, 109, 70, 30, 5, 253, 252, 250, 245, 236, 219, 191, 147, 91, 36, 6, 365

Offset: 1

Alford Arnold, Aug 25 2009

Sequence A164680 is not in the triangle since 45 = 3*15 and 15 is not a member of A000040 (the prime numbers).

			The desired triangle begins:
.1
.2..1
.5..4..2
.9..8..6..2
17.16.14..9..3
28.27.25.20.12..3
47.46.44.39.30.16..4
etc.
Note that 21 = 3*7 maps to 1,2,5,9,17,27,44,... A139672 is embedded in the triangle.

Cf. A008619 A002620 A006918 A097701 A139672 ... A000097 (embedded sequences).

A307373 Heinz numbers of integer partitions with at least three parts, the third of which is 2.

Original entry on oeis.org

27, 45, 54, 63, 75, 81, 90, 99, 105, 108, 117, 126, 135, 147, 150, 153, 162, 165, 171, 180, 189, 195, 198, 207, 210, 216, 225, 231, 234, 243, 252, 255, 261, 270, 273, 279, 285, 294, 297, 300, 306, 315, 324, 330, 333, 342, 345, 351, 357, 360, 363, 369, 378, 387

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A006918 (see Emeric Deutsch's comment there).

			The sequence of terms together with their prime indices begins:
   27: {2,2,2}
   45: {2,2,3}
   54: {1,2,2,2}
   63: {2,2,4}
   75: {2,3,3}
   81: {2,2,2,2}
   90: {1,2,2,3}
   99: {2,2,5}
  105: {2,3,4}
  108: {1,1,2,2,2}
  117: {2,2,6}
  126: {1,2,2,4}
  135: {2,2,2,3}
  147: {2,4,4}
  150: {1,2,3,3}
  153: {2,2,7}
  162: {1,2,2,2,2}
  165: {2,3,5}
  171: {2,2,8}
  180: {1,1,2,2,3}

Cf. A000726, A002620, A004250, A006918, A056239, A097701, A112798, A257990, A297113, A325164, A325169, A325170.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeOmega[#]>=3&&Reverse[primeMS[#]][[3]]==2&]

cp's OEIS Frontend

A008763 Expansion of g.f.: x^4/((1-x)*(1-x^2)^2*(1-x^3)).

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A115262 Correlation triangle for n+1.

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A002625 Expansion of 1/((1-x)^3*(1-x^2)^2*(1-x^3)).

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A139672 Convolution of A008619 and A001400.

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A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

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A164678 Convolve A008619 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

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A307373 Heinz numbers of integer partitions with at least three parts, the third of which is 2.

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Programs

Mathematica

A164679 Convolve A001399 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

A002625 Expansion of 1/((1-x)^3(1-x^2)^2(1-x^3)).