A006918 -id:A006918 - cp's OEIS frontend

A038163 G.f.: 1/((1-x)*(1-x^2))^3.

1, 3, 9, 19, 39, 69, 119, 189, 294, 434, 630, 882, 1218, 1638, 2178, 2838, 3663, 4653, 5863, 7293, 9009, 11011, 13377, 16107, 19292, 22932, 27132, 31892, 37332, 43452, 50388, 58140, 66861, 76551, 87381, 99351, 112651, 127281

Offset: 0

N. J. A. Sloane

Number of symmetric nonnegative integer 6 X 6 matrices with sum of elements equal to 4*n, under action of dihedral group D_4. - Vladeta Jovovic, May 14 2000
Equals the triangular sequence convolved with the aerated triangular sequence, [1, 0, 3, 0, 6, 0, 10, ...]. - Gary W. Adamson, Jun 11 2009
Number of partitions of n (n>=1) into 1s and 2s if there are three kinds of 1s and three kinds of 2s. Example: a(2)=9 because we have 11, 11', 11", 1'1', 1'1", 1"1", 2, 2', and 2". - Emeric Deutsch, Jun 26 2009
Equals the tetrahedral numbers with repeats convolved with the natural numbers: (1 + x + 4x^2 + 4x^3 + ...) * (1 + 2x + 3x^2 + 4x^3 + ...) = (1 + 3x + 9x^2 + 19x^3 + ...). - Gary W. Adamson, Dec 22 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,0,3,-1).

Cf. A008619, A006918, A001753.
Cf. A096338.
Column k=3 of A210391. - Alois P. Heinz, Mar 22 2012
Cf. A000217.

import Data.List (inits, intersperse)
a038163 n = a038163_list !! n
a038163_list = map
    (sum . zipWith (*) (intersperse 0 $ tail a000217_list) . reverse) $
    tail $ inits $ tail a000217_list where
-- Reinhard Zumkeller, Feb 27 2015

G := 1/((1-x)^3*(1-x^2)^3): Gser := series(G, x = 0, 42): seq(coeff(Gser, x, n), n = 0 .. 37); # Emeric Deutsch, Jun 26 2009
# alternative
A038163 := proc(n)
    (4*n^5+90*n^4+760*n^3+2970*n^2+5266*n+3285+(-1)^n*(30*n^2+270*n+555))/3840 ;
end proc:
seq(A038163(n),n=0..30) ; # R. J. Mathar, Feb 22 2021

CoefficientList[Series[1/((1-x)*(1-x^2))^3, {x, 0, 40}], x] (* Jean-François Alcover, Mar 11 2014 *)
LinearRecurrence[{3,0,-8,6,6,-8,0,3,-1},{1,3,9,19,39,69,119,189,294},50] (* Harvey P. Dale, Nov 24 2022 *)

a(2*k) = (4*k + 5)*binomial(k + 4, 4)/5 = A034263(k); a(2*k + 1) = binomial(k + 4, 4)*(15 + 4*k)/5 = A059599(k), k >= 0.
a(n) = (1/3840)*(4*n^5 + 90*n^4 + 760*n^3 + 2970*n^2 + 5266*n + 3285 + (-1)^n*(30*n^2 + 270*n + 555)). Recurrence: a(n) = 3*a(n-1) - 8*a(n-3) + 6*a(n-4) + 6*a(n-5) - 8*a(n-6) + 3*a(n-8) - a(n-9). - Vladeta Jovovic, Apr 24 2002
a(n+1) - a(n) = A096338(n+2). - R. J. Mathar, Nov 04 2008

A325164 Heinz numbers of integer partitions with Durfee square of length 2.

Original entry on oeis.org

9, 15, 18, 21, 25, 27, 30, 33, 35, 36, 39, 42, 45, 49, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 126, 129, 130, 132, 133, 135, 138, 140

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).
Also positions of 2 in A257990.
First differs from A105441 in lacking 125.
The Durfee length 1 case is A093641. The enumeration of Durfee length 2 partitions by sum is given by A006918, while that of Durfee length 3 partitions is given by A117485.

			The sequence of terms together with their prime indices begins:
   9: {2,2}
  15: {2,3}
  18: {1,2,2}
  21: {2,4}
  25: {3,3}
  27: {2,2,2}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  39: {2,6}
  42: {1,2,4}
  45: {2,2,3}
  49: {4,4}
  50: {1,3,3}
  51: {2,7}
  54: {1,2,2,2}
  55: {3,5}
  57: {2,8}
  60: {1,1,2,3}

Gus Wiseman, Young diagrams corresponding to the first 36 terms.

Cf. A006918, A056239, A093641, A112798, A115994, A117485, A252464, A257990, A325163, A325170.

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

A002624 Expansion of (1-x)^(-3) * (1-x^2)^(-2).

Original entry on oeis.org

1, 3, 8, 16, 30, 50, 80, 120, 175, 245, 336, 448, 588, 756, 960, 1200, 1485, 1815, 2200, 2640, 3146, 3718, 4368, 5096, 5915, 6825, 7840, 8960, 10200, 11560, 13056, 14688, 16473, 18411, 20520, 22800, 25270, 27930, 30800, 33880, 37191, 40733, 44528

Offset: 0

N. J. A. Sloane

Given an irregular triangular matrix M with the triangular numbers in every column shifted down twice for columns >0, A002624 = M * [1, 2, 3, ...]. Example: row 4 of triangle M = (15, 6, 1), then (15, 6, 1) dot (1, 2, 3) = a(4) = 30 = (15 + 12 + 3). - Gary W. Adamson, Mar 02 2010
The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums of A139600 are related to the sequence given above, e.g., Ze2(n) = a(n-1) - a(n-2) - a(n-3) + 4*a(n-4), with a(n) = 0 for n <= -1. For the definitions of these triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011
8*a(n) + 16*a(n+1) + 16*a(n+2) is the number of ways to place 3 queens on an (n+6) X (n+6) chessboard so that they diagonally attack each other exactly twice. Also true for the nonexistent terms for n=-1, n=-2 and n=-3 assuming that they are zeros. In graph-theory representation they thus form the corresponding open walk (Eulerian trail) with V={1,2,3} vertices and length of 2. - Antal Pinter, Dec 31 2015
a(n) is the number of partitions of n into parts with three kinds of 1 and two kinds of 2. - Joerg Arndt, Jan 18 2016

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..10000
Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
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]
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 20.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 204
Antal Pinter, Numerical solution of the k=3 Queens problem, 2011, Q(n) at p.8.
Antal Pinter, Software utility for enumerating positions of non-attacking and attacking chess pieces , Backtrack_V7Pro
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,-5,5,1,-3,1).

Cf. A047659, A139600, A180662.

[( (n+1)^4 +10*(n+1)^3 +32*(n+1)^2 +32*(n+1) +(6*(n+1) +15)*((n+1) mod 2) )/96 : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011

A002624:=-1/(z+1)**2/(z-1)**5; # Simon Plouffe in his 1992 dissertation

f[n_] := Block[{m = n - 1}, (m^4 + 10m^3 + 32m^2 + 32m + (6m + 15)Mod[m, 2])/96]; Table[ f[n], {n, 2, 45}]
(* Or *) CoefficientList[ Series[1/((1 - x)^3 (1 - x^2)^2), {x, 0, 44}], x] (* Robert G. Wilson v, Feb 26 2005 *)

Vec(1/(1-x)^3/(1-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Apr 19 2012

a(n)=(n^4 + 14*n^3 + 68*n^2 + 136*n - n%2*(6*n + 21))/96 + 1 \\ Charles R Greathouse IV, Feb 18 2016

a(n-1) = ( n^4 +10*n^3 +32*n^2 +32*n +(6*n +15)*(n mod 2) )/96.
From Antal Pinter, Oct 03 2014: (Start)
a(n) = C(n + 2, 2) + 2*C(n, 2) + 3*C(n - 2, 2) + 4*C(n - 4, 2) + ...
a(n) = Sum_{i = 1..z} i*C(n + 4 - 2i, 2)  where z = (2*n + 3 + (-1)^n)/4.
a(n) = (3*(2*n + 7)*(-1)^n + 2*n^4 + 28*n^3 + 136*n^2 + 266*n + 171)/192.
(End)
a(n) = A007009(n+1) - A001752(n-1) for n>0. - Antal Pinter, Dec 27 2015
a(n) = Sum_{j=0..n+1} A006918(j). - Richard Turk, Feb 18 2016

Formula and more terms from Frank Ellermann, Mar 14 2002

A000094 Number of trees of diameter 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 65, 88, 121, 161, 215, 280, 367, 471, 607, 771, 980, 1232, 1551, 1933, 2410, 2983, 3690, 4536, 5574, 6811, 8317, 10110, 12276, 14848, 17941, 21600, 25977, 31146, 37298, 44542, 53132, 63218, 75131, 89089

Offset: 1

N. J. A. Sloane

nonn

Number of partitions of n-1 with at least two parts of size 2 or larger. - Franklin T. Adams-Watters, Jan 13 2006
Also equal to the number of partitions p of n-1 such that max(p)-min(p) > 1. Example: a(7)=5 because we have [5,1],[4,2],[4,1,1],[3,2,1] and [3,1,1,1]. - Giovanni Resta, Feb 06 2006
Also number of partitions of n-1 with at least two parts that are smaller than the largest part. Example: a(7)=5 because we have [4,1,1],[3,2,1],[3,1,1,1],[2,2,1,1,1] and [2,1,1,1,1]. - Emeric Deutsch, May 01 2006
Also number of regions of n-1 that do not contain 1 as a part, n >= 2 (cf. A186114, A206437). - Omar E. Pol, Dec 01 2011
Also rank of the last region of n-1 multiplied by -1, n >= 2 (cf. A194447). - Omar E. Pol, Feb 11 2012
Also sum of ranks of the regions of n-1 that contain emergent parts, n >= 2 (cf. A182699). For the definition of "regions of n" see A206437. - Omar E. Pol, Feb 21 2012

			From _Gus Wiseman_, Apr 12 2019: (Start)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts of size 2 or larger, or non-hooks, are the following. The Heinz numbers of these partitions are given by A105441.
  (22)  (32)   (33)    (43)     (44)
        (221)  (42)    (52)     (53)
               (222)   (322)    (62)
               (321)   (331)    (332)
               (2211)  (421)    (422)
                       (2221)   (431)
                       (3211)   (521)
                       (22111)  (2222)
                                (3221)
                                (3311)
                                (4211)
                                (22211)
                                (32111)
                                (221111)
The a(5) = 1 through a(9) = 14 partitions of n-1 whose maximum part minus minimum part is at least 2 are the following. The Heinz numbers of these partitions are given by A307516.
  (31)  (41)   (42)    (52)     (53)
        (311)  (51)    (61)     (62)
               (321)   (331)    (71)
               (411)   (421)    (422)
               (3111)  (511)    (431)
                       (3211)   (521)
                       (4111)   (611)
                       (31111)  (3221)
                                (3311)
                                (4211)
                                (5111)
                                (32111)
                                (41111)
                                (311111)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts that are smaller than the largest part are the following. The Heinz numbers of these partitions are given by A307517.
  (211)  (311)   (321)    (322)     (422)
         (2111)  (411)    (421)     (431)
                 (2211)   (511)     (521)
                 (3111)   (3211)    (611)
                 (21111)  (4111)    (3221)
                          (22111)   (3311)
                          (31111)   (4211)
                          (211111)  (5111)
                                    (22211)
                                    (32111)
                                    (41111)
                                    (221111)
                                    (311111)
                                    (2111111)
(End)

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).

Christian G. Bower, Table of n, a(n) for n = 1..500
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Miloslav Znojil, Perturbation theory near degenerate exceptional points, arXiv:2008.00479 [math-ph], 2020.
Index entries for sequences related to trees

Cf. A000041, A206437, A034853, A000147 (diameter 5).

Cf. A006918, A083751, A084835, A105441, A115720, A257990, A307516, A307517, A325164.

g:=x/product(1-x^j,j=1..70)-x-x^2/(1-x)^2: gser:=series(g,x=0,48): seq(coeff(gser,x,n),n=1..46); # Emeric Deutsch, May 01 2006
A000094 := proc(n)
    combinat[numbpart](n-1)-n+1 ;
end proc: # R. J. Mathar, May 17 2016

t=Table[PartitionsP[n]-n,{n,0,45}];
ReplacePart[t,0,1]
(* Clark Kimberling, Mar 05 2012 *)
CoefficientList[1/QPochhammer[x]-x/(1-x)^2-1+O[x]^50, x] (* Jean-François Alcover, Feb 04 2016 *)

a(n+1) = A000041(n)-n for n>0. - John W. Layman
G.f.: x/product(1-x^j,j=1..infinity)-x-x^2/(1-x)^2. - Emeric Deutsch, May 01 2006
G.f.: sum(sum(x^(i+j+1)/product(1-x^k, k=i..j), i=1..j-2), j=3..infinity). - Emeric Deutsch, May 01 2006
a(n+1) = Sum_{m=1..n} A083751(m). - Gregory Gerard Wojnar, Oct 13 2020

More terms from Franklin T. Adams-Watters, Jan 13 2006

A011847 Triangle of numbers read by rows: T(n,k) = floor( C(n,k)/(k+1) ), where k=0..n-1 and n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 3, 7, 8, 7, 3, 1, 1, 4, 9, 14, 14, 9, 4, 1, 1, 4, 12, 21, 25, 21, 12, 4, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 5, 18, 41, 66, 77, 66, 41, 18, 5, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1, 6, 26, 71, 143, 214, 245, 214, 143, 71, 26, 6, 1

Offset: 1

N. J. A. Sloane

When k+1 is a prime >= 2, then T(n,k) = floor(C(n,k)/(k+1)) is the number of aperiodic necklaces of n+1 beads of 2 colors such that k+1 of them are black and n-k of them are white. This is not true when k+1 is a composite >= 4. For more details, see the comments for sequences A032168 and A032169. - Petros Hadjicostas, Aug 27 2018

Differs from A245558 from row n = 9, k = 4 on. - M. F. Hasler, Sep 29 2018

			Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   1;
  1, 2,  3,   2,   1;
  1, 3,  5,   5,   3,   1;
  1, 3,  7,   8,   7,   3,    1;
  1, 4,  9,  14,  14,   9,    4,    1;
  1, 4, 12,  21,  25,  21,   12,    4,    1;
  1, 5, 15,  30,  42,  42,   30,   15,    5,    1;
  1, 5, 18,  41,  66,  77,   66,   41,   18,    5,   1;
  1, 6, 22,  55,  99, 132,  132,   99,   55,   22,   6,   1;
  1, 6, 26,  71, 143, 214,  245,  214,  143,   71,  26,   6,   1;
  1, 7, 30,  91, 200, 333,  429,  429,  333,  200,  91,  30,   7,  1;
  1, 7, 35, 113, 273, 500,  715,  804,  715,  500, 273, 113,  35,  7, 1;
  1, 8, 40, 140, 364, 728, 1144, 1430, 1430, 1144, 728, 364, 140, 40, 8, 1;
...
More than the usual number of rows are shown in order to distinguish this triangle from A245558, from which it differs in rows 9, 11, 13, ....
From _Petros Hadjicostas_, Aug 27 2018: (Start)
For k+1 = 2 and n >= k+1 = 2, the n-th element of column k=1 above, [0, 1, 1, 2, 2, 3, 3, 4, 4, ...] (i.e., the number A008619(n-2) = floor(n/2)), gives the number of aperiodic necklaces of n+1 beads of 2 colors such that 2 of them are black and n-1 of them are white. (The offset of sequence A008619 is 0.)
For k+1 = 3 and n >= k+1 = 3, the n-th element of column k=2 above, [0, 0, 1, 2, 3, 5, 7, 9, 12, ...] (i.e., the number A001840(n-2) = floor(C(n,2)/3)), gives the number of aperiodic necklaces of n+1 beads of 2 colors such that 3 of them are black and n-2 of them are white. (The offset of sequence A001840 is 0.)
For k+1 = 5 and n >= k+1 = 5, the n-th element of column k=4 above, [0, 0, 0, 0, 1, 3, 7, 14, 25, 42, ... ] (i.e., the number A011795(n) = floor(C(n,4)/5)), gives the number of aperiodic necklaces of n+1 beads of 2 colors such that 5 of them are black and n-4 of them are white. (The offset of sequence A011795 is 0.)
Counterexample for k+1 = 4: It can be proved that, for n >= k+1 = 4, the number of aperiodic necklaces of n+1 beads of 2 colors such that 4 of them are black and n-3 of them are white is (1/4)*Sum_{d|4} mu(d)*I(d|n+1)*C(floor((n+1)/d) - 1, 4/d - 1) = (1/4)*(C(n, 3) - I(2|n+1)*floor((n-1)/2)), where I(a|b) = 1 if integer a divides integer b, and 0 otherwise. For n odd >= 9, the number (1/4)*(C(n, 3) - I(2|n+1)*floor((n-1)/2)) = A006918(n-3) is not equal to floor(C(n,3)/4) = A011842(n).
(End)

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Columns include A008619, A001840, A011842, A011795, A011843, A011797, A011844, A011845, A011846, A032169.
Sums: A095718 (row), A095719 (diagonal).
Cf. A006918, A245558.

A011847:= func< n,k | Floor(Binomial(n+1,k+1)/(n+1)) >;
[A011847(n,k): k in [0..n-1], n in [1..20]]; // G. C. Greubel, Oct 20 2024

Table[Floor[Binomial[n,k]/(k+1)],{n,20},{k,0,n-1}]//Flatten (* Harvey P. Dale, Jan 09 2019 *)

A011847(n,k)=binomial(n,k)\(k+1) \\ M. F. Hasler, Sep 30 2018

def A011847(n,k): return binomial(n+1,k+1)//(n+1)
flatten([[A011847(n,k) for k in range(n)] for n in range(1,21)]) # G. C. Greubel, Oct 20 2024

The rows are palindromic: T(n, k) = T(n, n-k-1) for n >= 1 and 0 <= k <= n-1.

Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A095719(n). - G. C. Greubel, Oct 20 2024

A232432 Number of compositions of n avoiding the pattern 111.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 21, 34, 59, 114, 178, 284, 522, 823, 1352, 2133, 3739, 5807, 9063, 14074, 23639, 36006, 56914, 87296, 131142, 214933, 324644, 487659, 739291, 1108457, 1724673, 2558386, 3879335, 5772348, 8471344, 12413666, 19109304, 27886339, 40816496

Offset: 0

Alois P. Heinz, Nov 23 2013

nonn

Number of compositions of n into parts with multiplicity <= 2.

			a(4) = 7: [4], [3,1], [2,2], [1,3], [2,1,1], [1,2,1], [1,1,2].
a(5) = 11: [5], [4,1], [3,2], [2,3], [1,4], [3,1,1], [2,2,1], [1,3,1], [2,1,2], [1,2,2], [1,1,3].
a(6) = 21: [6], [4,2], [3,3], [5,1], [2,4], [1,5], [2,1,3], [1,2,3], [1,1,4], [4,1,1], [3,2,1], [2,3,1], [1,4,1], [3,1,2], [1,3,2], [1,2,2,1], [2,1,1,2], [1,2,1,2], [1,1,2,2], [2,2,1,1], [2,1,2,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A000726 (partitions avoiding 111), A032020 (pattern 11), A128695 (adjacent pattern 111).
Column k=2 of A243081.
The case of partitions is ranked by A004709.
The version for patterns is A080599.
(1,1,1,1)-avoiding partitions are counted by A232464.
The (1,1,1)-matching version is A335455.
Patterns matched by compositions are counted by A335456.
The version for prime indices is A335511.
(1,1,1)-avoiding compositions are ranked by A335513.
Cf. A011782, A006918, A102726, A106356, A238279, A333755.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j)/j!, j=0..min(n/i, 2))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);

f[list_]:=Apply[And,Table[Count[list,i]<3,{i,1,Max[list]}]];
g[list_]:=Length[list]!/Apply[Times,Table[Count[list,i]!,{i,1,Max[list]}]];
a[n_] := If[n == 0, 1, Total[Map[g, Select[IntegerPartitions[n], f]]]];
Table[a[n], {n, 0, 35}] (* Geoffrey Critzer, Nov 25 2013, updated by Jean-François Alcover, Nov 20 2023 *)

A062109 Expansion of ((1-x)/(1-2*x))^4.

Original entry on oeis.org

1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, 39680, 96256, 230656, 546816, 1284096, 2990080, 6909952, 15859712, 36175872, 82051072, 185139200, 415760384, 929562624, 2069889024, 4591714304, 10150215680, 22364028928, 49123688448, 107592286208, 235015241728, 512040632320

Offset: 0

Henry Bottomley, May 30 2001

If X_1,X_2,...,X_n are 2-blocks of a (2n+4)-set X then, for n >= 1, a(n+1) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 23 2007
If the offset here is set to zero, the binomial transform of A006918. - R. J. Mathar, Jun 29 2009
a(n) is the number of weak compositions of n with exactly 3 parts equal to 0. - Milan Janjic, Jun 27 2010
Binomial transform of A002623. - Carl Najafi, Jan 22 2013
Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^4; see A291000. - Clark Kimberling, Aug 24 2017

Harry J. Smith, Table of n, a(n) for n = 0..200
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Milan Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^4)); // G. C. Greubel, Oct 16 2018

seq(coeff(series(((1-x)/(1-2*x))^4, x,n+1),x,n),n=0..30); # Muniru A Asiru, Jul 01 2018

CoefficientList[Series[(1 - x)^4/(1 - 2 x)^4, {x, 0, 26}], x] (* Michael De Vlieger, Jul 01 2018 *)
LinearRecurrence[{8,-24,32,-16},{1,4,14,44,129},30] (* Harvey P. Dale, Sep 02 2022 *)

a(n)=if(n<1,n==0,(n+5)*(n^2+13*n+18)*2^n/96)

a(n) = (n+5)*(n^2 + 13*n + 18)*2^(n-5)/3, with a(0)=1.
a(n) = A055809(n-5)*2^(n-4).
a(n) = 2*a(n-1) + A058396(n) - A058396(n-1).
a(n) = Sum_{kA058396(n).
a(n) = A062110(4, n).
G.f.: (1-x)^4/(1-2*x)^4.

A242093 Number A(n,k) of inequivalent n X k binary matrices, where equivalence means permutations of rows or columns or the symbol set; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 3, 8, 8, 3, 1, 1, 3, 14, 18, 14, 3, 1, 1, 4, 20, 47, 47, 20, 4, 1, 1, 4, 30, 95, 173, 95, 30, 4, 1, 1, 5, 40, 200, 545, 545, 200, 40, 5, 1, 1, 5, 55, 367, 1682, 2812, 1682, 367, 55, 5, 1, 1, 6, 70, 674, 4745, 14386, 14386, 4745, 674, 70, 6, 1

Offset: 0

Alois P. Heinz, Aug 14 2014

			A(1,4) = 3: [0 0 0 0], [1 0 0 0], [1 1 0 0].
A(1,5) = 3: [0 0 0 0 0], [1 0 0 0 0], [1 1 0 0 0].
A(2,2) = 5:
  [0 0]  [1 0]  [1 1]  [1 0]  [1 0]
  [0 0], [0 0], [0 0], [1 0], [0 1].
A(3,2) = 8:
  [0 0]  [1 0]  [1 1]  [1 0]  [1 0]  [1 0]  [1 0]  [1 1]
  [0 0], [0 0], [0 0], [1 0], [0 1], [1 0], [0 1], [1 0].
  [0 0]  [0 0]  [0 0]  [0 0]  [0 0]  [1 0]  [1 0]  [0 0]
Square array A(n,k) begins:
  1, 1,  1,   1,    1,     1,       1,        1, ...
  1, 1,  2,   2,    3,     3,       4,        4, ...
  1, 2,  5,   8,   14,    20,      30,       40, ...
  1, 2,  8,  18,   47,    95,     200,      367, ...
  1, 3, 14,  47,  173,   545,    1682,     4745, ...
  1, 3, 20,  95,  545,  2812,   14386,    68379, ...
  1, 4, 30, 200, 1682, 14386,  126446,  1072086, ...
  1, 4, 40, 367, 4745, 68379, 1072086, 16821330, ...

Columns (or rows) k=0-10 give: A000012, A008619, A006918(n+1), A246148, A246149, A246150, A246151, A246152, A246153, A246154, A246155.
Main diagonal gives A091059.
Cf. A028657, A241956, A242095.

with(numtheory):
b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
g:= proc(n, k) option remember; add(add(add(mul(mul(add(d*
      coeff(u, x, d), d=divisors(ilcm(i, j)))^(igcd(i, j)*
      coeff(s, x, i)*coeff(t, x, j)), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(u, x, i)*coeff(u, x, i)!,
      i=1..degree(u))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s)), u=b(2$2)), t=b(n$2)), s=b(k$2))
    end:
A:= (n, k)-> g(sort([n, k])[]):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten[Table[Map[ Function[p, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]]];
g[n_, k_] := g[n, k] = Sum[Sum[Sum[Product[Product[With[{gc = GCD[i, j]* Coefficient[s, x, i]*Coefficient[t, x, j]}, If[gc == 0, 1, Sum[d* Coefficient[u, x, d], {d, Divisors[LCM[i, j]]}]^gc]], {j, 1, Exponent[t, x]}],
{i, Exponent[s, x]}]/Product[i^Coefficient[u, x, i]*Coefficient[u, x, i]!,
{i, Exponent[u, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!,
{i, Exponent[t, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!,
{i, Exponent[s, x]}], {u, b[2, 2]}], {t, b[n, n]}], {s, b[k, k]}];
A[n_, k_] := g @@ Sort[{n, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 25 2016, adapted from Maple, updated Jan 01 2021 *)

A096338 a(n) = (2/(n-1))a(n-1) + ((n+5)/(n-1))a(n-2) with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 2, 6, 10, 20, 30, 50, 70, 105, 140, 196, 252, 336, 420, 540, 660, 825, 990, 1210, 1430, 1716, 2002, 2366, 2730, 3185, 3640, 4200, 4760, 5440, 6120, 6936, 7752, 8721, 9690, 10830, 11970, 13300, 14630, 16170, 17710, 19481, 21252, 23276, 25300, 27600

Offset: 0

Benoit Cloitre, Jun 28 2004

Without the leading zero, Poincaré series [or Poincare series] P(C_{2,2}; t).
Starting (1, 2, 6, ...) = partial sums of the tetrahedral numbers, A000292 with repeats: (1, 1, 4, 4, 10, 10, 20, 20, 35, 35, ...). - Gary W. Adamson, Mar 30 2009
Starting with 1 = [1, 2, 3, ...] convolved with the aerated triangular series, [1, 0, 3, 0, 6, ...]. - Gary W. Adamson, Jun 11 2009
From Alford Arnold, Oct 14 2009: (Start)
a(n) is also related to Dyck Paths. Note that
0 1 2 6 10 20 30 50 70 105 ...
minus
0 0 0 0 1 2 6 10 20 30 ...
equals
0 1 2 6 9 18 24 40 50 75 ... A028724
(End)
The Kn11, Kn12, Kn13, Fi1 and Ze1 triangle sums of A139600 are related to the sequence given above; e.g., Ze1(n) = 3*A096338(n-1) - 3*A096338(n-3) + 9*A096338(n-4), with A096338(n) = 0 for n <= -1. For the definition of these triangle sums, see A180662. - Johannes W. Meijer, Apr 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dragomir Z. Djokovic, Poincaré series [or Poincare series] of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 3.
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4, 19.
M. Navascues and T. Vertesi, The Structure of Matrix Product States, arXiv preprint arXiv:1509.04507 [quant-ph], 2015-2018.
Miguel Navascués and Tamás Vértesi, Bond dimension witnesses and the structure of homogeneous matrix product states, Quantum 2 (2018): 50.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A006918, A038163, A060099, A058187, A000292, A028724.

A096338:=n->-(floor(n/2)+1)*(floor(n/2)+2)*(floor(n/2)+3)*(3*floor(n/2)-2*n)/12; seq(A096338(k),k=0..100); # Wesley Ivan Hurt, Oct 04 2013

t = {0, 1}; Do[AppendTo[t, (2/(n - 1))*t[[-1]] + ((n + 5)/(n - 1))*t[[-2]]], {n, 2, 50}]; t (* T. D. Noe, Oct 08 2013 *)
CoefficientList[Series[x/((1 - x)^2*(1 - x^2)^3), {x, 0, 45}], x] (* or *)
Nest[Append[#1, (2/(#2 - 1))*#1[[-1]] + ((#2 + 5)/(#2 - 1))*#1[[-2]] ] & @@ {#, Length@ #} &, {0, 1}, 44] (* Michael De Vlieger, May 30 2018 *)

G.f.: x/((1-x)^2*(1-x^2)^3). - Ralf Stephan, Jun 29 2004
a(n) = Sum_{k=1..floor(n/2)+1} ( Sum_{i=1..k} i*(n-2*k+2) ) = -(floor(n/2)+1) * (floor(n/2)+2) * (floor(n/2)+3) * (3*floor(n/2) - 2*n)/12. - Wesley Ivan Hurt, Sep 26 2013
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8). - Wesley Ivan Hurt, Nov 26 2020
128*a(n) = 8*n^3 +94/3*n^2 +44*n +15 +2/3*n^4 -2*(-1)^n*n^2 -12*(-1)^n*n -15*(-1)^n. - R. J. Mathar, Mar 23 2021

A122196 Fractal sequence: count down by 2's from successive integers.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 5, 3, 1, 6, 4, 2, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 10, 8, 6, 4, 2, 11, 9, 7, 5, 3, 1, 12, 10, 8, 6, 4, 2, 13, 11, 9, 7, 5, 3, 1, 14, 12, 10, 8, 6, 4, 2, 15, 13, 11, 9, 7, 5, 3, 1, 16, 14, 12, 10, 8, 6, 4, 2, 17, 15, 13, 11, 9, 7, 5, 3, 1, 18, 16, 14, 12, 10, 8, 6, 4, 2, 19, 17

Offset: 1

Franklin T. Adams-Watters, Aug 25 2006

First differences of A076644. Fractal - deleting the first occurrence of each integer leaves the original sequence. Also, original sequence plus 1. 1's occur at square indices. New values occur at indices m^2+1 and m^2+m+1.
Ordinal transform of A122197.
Row sums give A002620. - Gary W. Adamson, Nov 29 2008
From Gary W. Adamson, Dec 05 2009: (Start)
A122196 considered as an infinite lower triangular matrix * [1,2,3,...] =
A006918 starting (1, 2, 5, 8, 14, 20, 30, 40, ...).
Let A122196 = an infinite lower triangular matrix M; then lim_{n->infinity} M^n = A171238, a left-shifted vector considered as a matrix. (End)
A122196 is the fractal sequence associated with the dispersion A082156; that is, A122196(n) is the number of the row of A082156 that contains n. - Clark Kimberling, Aug 12 2011
From Johannes W. Meijer, Sep 09 2013: (Start)
The alternating row sums lead to A004524(n+2).
The antidiagonal sums equal A001840(n). (End)

			The first few rows of the sequence a(n) as a triangle T(n, k):
n/k  1   2   3
1    1
2    2
3    3,  1
4    4,  2
5    5,  3,  1
6    6,  4,  2

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A076644, A122197, A000290, A033638, A002620, A006918, A171238, A082156, A000267.

a122196 n = a122196_list !! (n-1)
a122196_list = concatMap (\x -> enumFromThenTo x (x - 2) 1) [1..]
-- Reinhard Zumkeller, Jul 19 2012

From Johannes W. Meijer, Sep 09 2013: (Start)
a := proc(n) local t: t:=floor((sqrt(4*n-3)-1)/2): floor(sqrt(4*n-1))-2*((n-1) mod (t+1)) end: seq(a(n), n=1..92); # End first program.
T := (n, k) -> n-2*k+2: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..18); # End second program. (End)

Flatten@Range[Range[10], 1, -2] (* Birkas Gyorgy, Apr 07 2011 *)

From Boris Putievskiy, Sep 09 2013: (Start)
a(n) = 2*(1-A122197(n)) + A000267(n-1).
a(n) = floor(sqrt(4*n-1)) - 2*((n-1) mod (t+1)), where t = floor((sqrt(4*n-3)-1)/2). (End)
From Johannes W. Meijer, Sep 09 2013: (Start)
T(n, k) = n - 2*k + 2, for n >= 1 and 1 <= k <= floor((n+1)/2).
T(n, k) = A002260(n, n-2*k+2). (End)

cp's OEIS Frontend

A038163 G.f.: 1/((1-x)*(1-x^2))^3.

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A325164 Heinz numbers of integer partitions with Durfee square of length 2.

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A002624 Expansion of (1-x)^(-3) * (1-x^2)^(-2).

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A000094 Number of trees of diameter 4.

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A011847 Triangle of numbers read by rows: T(n,k) = floor( C(n,k)/(k+1) ), where k=0..n-1 and n >= 1.

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A232432 Number of compositions of n avoiding the pattern 111.

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A062109 Expansion of ((1-x)/(1-2*x))^4.

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A096338 a(n) = (2/(n-1))a(n-1) + ((n+5)/(n-1))a(n-2) with a(0)=0 and a(1)=1.