A077043 -id:A077043 - cp's OEIS frontend

A077042 Square array read by falling antidiagonals of central polynomial coefficients: largest coefficient in expansion of (1 + x + x^2 + ... + x^(n-1))^k = ((1-x^n)/(1-x))^k, i.e., the coefficient of x^floor(k(n-1)/2) and of x^ceiling(k(n-1)/2); also number of compositions of floor(k*(n+1)/2) into exactly k positive integers each no more than n.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 6, 7, 4, 1, 1, 0, 1, 10, 19, 12, 5, 1, 1, 0, 1, 20, 51, 44, 19, 6, 1, 1, 0, 1, 35, 141, 155, 85, 27, 7, 1, 1, 0, 1, 70, 393, 580, 381, 146, 37, 8, 1, 1, 0, 1, 126, 1107, 2128, 1751, 780, 231, 48, 9, 1, 1, 0, 1, 252, 3139

Offset: 0

Henry Bottomley, Oct 22 2002

From Michel Marcus, Dec 01 2012: (Start)
A pair of numbers written in base n are said to be comparable if all digits of the first number are at least as big as the corresponding digit of the second number, or vice versa. Otherwise, this pair will be defined as uncomparable. A set of pairwise uncomparable integers will be called anti-hierarchic.
T(n,k) is the size of the maximal anti-hierarchic set of integers written with k digits in base n.
For example, for base n=2 and k=4 digits:
- 0 (0000) and 15 (1111) are comparable, while 6 (0110) and 9 (1001) are uncomparable,
- the maximal antihierarchic set is {3 (0011), 5 (0101), 6 (0110), 9 (1001), 10 (1010), 12 (1100)} with 6 elements that are all pairwise uncomparable. (End)

			Rows of square array start:
  1,    0,    0,    0,    0,    0,    0, ...
  1,    1,    1,    1,    1,    1,    1, ...
  1,    1,    2,    3,    6,   10,   20, ...
  1,    1,    3,    7,   19,   51,  141, ...
  1,    1,    4,   12,   44,  155,  580, ...
  1,    1,    5,   19,   85,  381, 1751, ...
  ...
Read by antidiagonals:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1, 1;
  0, 1, 3, 3, 1, 1;
  0, 1, 6, 7, 4, 1, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Denis Bouyssou, Thierry Marchant, Marc Pirlot, The size of the largest antichains in products of linear orders, arXiv:1903.07569 [math.CO], 2019.
J. W. Sander, On maximal antihierarchic sets of integers, Discrete Mathematics, Volume 113, Issues 1-3, 5 April 1993, Pages 179-189.
Index entries for sequences related to compositions

Rows include A000007, A000012, A001405, A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015, A201549, A225779, A201550. Columns include A000012, A000012, A001477, A077043, A005900, A077044, A071816, A133458.
Central diagonal is A077045, with A077046 and A077047 either side. Cf. A067059, A270918, A201552.
Cf. A273975.

t[n_, k_] := Max[ CoefficientList[ Series[ ((1-x^n) / (1-x))^k, {x, 0, k*(n-1)}], x]]; t[0, 0] = 1; t[0, ] = 0; Flatten[ Table[ t[n-k, k], {n, 0, 12}, {k, n, 0, -1}]] (* _Jean-François Alcover, Nov 04 2011 *)

T(n,k)=if(n<1 || k<1,k==0,vecmax(Vec(((1-x^n)/(1-x))^k)))

By the central limit theorem, T(n,k) is roughly n^(k-1)*sqrt(6/(Pi*k)).

T(n,k) = Sum{j=0,h/n} (-1)^j*binomial(k,j)*binomial(k-1+h-n*j,k-1) with h=floor(k*(n-1)/2), k>0. - Michel Marcus, Dec 01 2012

A274581 Number T(n,k) of set partitions of [n] with alternating parity of elements and exactly k blocks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 5, 7, 4, 1, 0, 1, 7, 14, 12, 5, 1, 0, 1, 11, 30, 33, 19, 6, 1, 0, 1, 15, 57, 84, 62, 27, 7, 1, 0, 1, 23, 119, 222, 204, 108, 37, 8, 1, 0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1, 0, 1, 47, 460, 1425, 2006, 1558, 763, 254, 61, 10, 1

Offset: 0

Alois P. Heinz, Jun 29 2016

			T(5,1) = 1: 12345.
T(5,2) = 5: 1234|5, 123|45, 12|345, 145|23, 1|2345.
T(5,3) = 7: 123|4|5, 12|34|5, 12|3|45, 1|234|5, 145|2|3, 1|2|345, 1|23|45.
T(5,4) = 4: 12|3|4|5, 1|23|4|5, 1|2|34|5, 1|2|3|45.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  3,   3,   1;
  0, 1,  5,   7,   4,   1;
  0, 1,  7,  14,  12,   5,   1;
  0, 1, 11,  30,  33,  19,   6,   1;
  0, 1, 15,  57,  84,  62,  27,   7,  1;
  0, 1, 23, 119, 222, 204, 108,  37,  8, 1;
  0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..35, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000007, A057427, A052955(n-2) for n>1, A305777, A305778, A305779, A305780, A305781, A305782, A305783, A305784.
Diagonals include A000012, A001477, A077043.
Row sums give A274547.
T(n,ceiling(n/2)) gives A305785.
Cf. A124419, A274310 (parities alternate within blocks), A305823.

b:= proc(l, i, t) option remember; `if`(l=[], x,
     `if`(l[1]=t, 0, expand(x*b(subsop(1=[][], l), 1, 1-t)
       ))+add(`if`(l[j]=t, 0, b(subsop(j=[][], l), j, 1-t)
       ), j=i..nops(l)))
    end:
T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, j), j=0..n))(
         b([seq(irem(i, 2), i=2..n)], 1$2))):
seq(T(n), n=0..12);

b[l_, i_, t_] := b[l, i, t] = If[l == {}, x, If[l[[1]] == t, 0, Expand[x*b[Rest[l], 1, 1 - t]]] + Sum[If[l[[j]] == t, 0, b[Delete[l, j], j, 1 - t]], {j, i, Length[l]}]];
T[n_] := If[n==0, {1}, Function[p, Table[Coefficient[p, x, j], {j, 0, n}]][ b[Table[Mod[i, 2], {i, 2, n}], 1, 1]]];
Flatten[Table[T[n], {n, 0, 12}]] (* Jean-François Alcover, May 27 2018, from Maple *)

Sum_{k=0..n} k * T(n,k) = A305823(n).

A132894 Number of (1,0) steps in all paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., in all length-n left factors of Motzkin paths).

Original entry on oeis.org

0, 1, 4, 15, 52, 175, 576, 1869, 6000, 19107, 60460, 190333, 596652, 1863745, 5804176, 18028755, 55873872, 172818243, 533589660, 1644921789, 5063762220, 15568666029, 47811348816, 146675181975, 449538774048, 1376564658525

Offset: 0

Emeric Deutsch, Oct 07 2007

nonn

Number of peaks (i.e., UDs) in all paths of length n+1 with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., in all length n+1 left factors of Motzkin paths). Example: a(2)=4 because in the 13 (=A005773(4)) length-3 left factors of Motzkin paths, namely HHH, HHU, H(UD), HUH, HUU, (UD)H, (UD)U, UHD, UHH, UHU, U(UD), UUH and UUU, we have altogether 4 peaks (shown between parentheses).

This could be called the Motzkin transform of A077043 because the substitution x -> x*A001006(x) in the independent variable of the g.f. of A077043 yields the g.f. of this sequence here. - R. J. Mathar, Nov 10 2008

			a(2) = 4 because in the 5 (=A005773(3)) length-2 left factors of Motzkin paths, namely HH, HU, UD, UH and UU, we have altogether 4 H steps.
G.f. = x + 4*x^2 + 15*x^3 + 52*x^4 + 175*x^5 + 576*x^6 + 1869*x^7 + 6000*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. Asinowski and G. Rote, Point sets with many non-crossing matchings, arXiv preprint arXiv:1502.04925 [cs.CG], 2015.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv preprint arXiv:1203.6792, 2012; and also, J. Int. Seq. 17 (2014) #14.1.5.
Mark Shattuck, Subword Patterns in Smooth Words, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 6.

Cf. A005773, A107230, A132893.

Column k=1 of A328347.

a := n -> add(k*binomial(n, k)*binomial(n-k, floor((n-k)/2)), k=0..n): seq(a(n), n=0..25);
# second Maple program:
a:= proc(n) a(n):=`if`(n<2, n, 2*n/(n-1)*a(n-1)+3*a(n-2)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 15 2013

a[n_] := n*Hypergeometric2F1[3/2, 1-n, 2, 4]; Table[ a[n] // Abs, {n, 0, 25}] (* Jean-François Alcover, Jul 10 2013 *)
a[ n_] := If[ n < 0, 0, -(-1)^n n Hypergeometric2F1[ 3/2, 1 - n, 2, 4]]; (* Michael Somos, Aug 06 2014 *)

A132894 = lambda n: (-1)^(n+1)*jacobi_P(n-1,1,-n+1/2,-7)
[Integer(A132894(n).n(40),16) for n in range(26)] # Peter Luschny, Sep 23 2014

a(n) = Sum_{k=0..n} k*A107230(n,k).
a(n) = Sum_{k=0..floor((n+1)/2)} k*A132893(n+1,k).
a(n) = Sum_{k=0..n} k*C(n,k)*C(n-k, floor((n-k)/2)).
G.f.: z/((1-3*z)*sqrt(1-2*z-3*z^2)).
a(n) = Sum_{k=0..n} k*C(n,k)*C(2*k,k)*(-1)^(n-k). - Wadim Zudilin, Oct 11 2010
E.g.f.: exp(x)*x*(BesselI(0, 2*x) + BesselI(1, 2*x)). - Peter Luschny, Aug 25 2012
a(n) = 2*n/(n-1)*a(n-1) + 3*a(n-2) for n>=2, a(n) = n for n<2. a(n) = n*A005773(n). - Alois P. Heinz, Jul 15 2013
a(n) ~ 3^(n-1/2)*sqrt(n/Pi). - Vaclav Kotesovec, Oct 08 2013
a(n) = (-1)^(n+1)*JacobiP(n-1,1,-n+1/2,-7). - Peter Luschny, Sep 23 2014

A235988 Sum of the partition parts of 3n into 3 parts.

Original entry on oeis.org

3, 18, 63, 144, 285, 486, 777, 1152, 1647, 2250, 3003, 3888, 4953, 6174, 7605, 9216, 11067, 13122, 15447, 18000, 20853, 23958, 27393, 31104, 35175, 39546, 44307, 49392, 54897, 60750, 67053, 73728, 80883, 88434, 96495, 104976, 113997, 123462, 133497, 144000

Offset: 1

Wesley Ivan Hurt, Jan 17 2014

			a(2) = 18; 3(2) = 6 has 3 partitions into 3 parts: (4, 1, 1), (3, 2, 1), and (2, 2, 2). The sum of the parts is 18.
Figure 1: The partitions of 3n into 3 parts for n = 1, 2, 3, ...
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..    3n
------------------------------------------------------------------------
    3           18          63         144         285      ..   a(n)
- _Wesley Ivan Hurt_, Sep 07 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A001651, A077043, A093353.

[3*n^3-3*n*Floor(n^2/4): n in [1..100]]; // Wesley Ivan Hurt, Nov 01 2015

[3*n*(1-(-1)^n+6*n^2)/8: n in [1..40]]; // Vincenzo Librandi, Nov 18 2015

A235988:=n->3*n^3 - 3*n*floor(n^2/4); seq(A235988(n), n=1..100);

Table[3 n^3 - 3 n*Floor[n^2/4], {n, 100}] (* or *) CoefficientList[ Series[3*x*(x^4 + 4*x^3 + 8*x^2 + 4*x + 1)/((x - 1)^4*(x + 1)^2), {x, 0, 30}], x]
LinearRecurrence[{2,1,-4,1,2,-1},{3,18,63,144,285,486},40] (* Harvey P. Dale, May 17 2018 *)

a(n)=3*n^3 - n^2\4*3*n \\ Charles R Greathouse IV, Oct 07 2015

x='x+O('x^50); Vec(3*x*(x^4+4*x^3+8*x^2+4*x+1)/((x-1)^4*(x+1)^2)) \\ Altug Alkan, Nov 01 2015

a(n) = 3*n^3 - 3*n*floor(n^2/4).
a(n) = 3n * A077043(n).
a(n) = a(n-1) + 3*A077043(n-1) + A001651(n) + A093353(3n-2).
From Colin Barker, Jan 18 2014: (Start)
a(n) = (3*n*(1-(-1)^n+6*n^2))/8.
G.f.: 3*x*(x^4+4*x^3+8*x^2+4*x+1) / ((x-1)^4*(x+1)^2). (End)
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 6. - Wesley Ivan Hurt, Nov 15 2015
E.g.f.: 3*x*((4 + 9*x + 3*x^2)*cosh(x) + 3*(1 + 3*x + x^2)*sinh(x))/4. - Stefano Spezia, Feb 09 2023

a(165) in b-file corrected by Andrew Howroyd, Feb 21 2018

A008747 Expansion of (1+x^4)/((1-x)(1-x^2)(1-x^3)).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 14, 17, 21, 24, 29, 33, 38, 43, 49, 54, 61, 67, 74, 81, 89, 96, 105, 113, 122, 131, 141, 150, 161, 171, 182, 193, 205, 216, 229, 241, 254, 267, 281, 294, 309, 323, 338, 353, 369, 384, 401, 417, 434, 451, 469, 486, 505, 523, 542, 561

Offset: 0

N. J. A. Sloane

For n>=1, the set {A008747(6n+-1)} is the set of numbers of the form a^2 + 5*(a+1)^2 for -inf < a < inf. Furthermore the set A008747(6n) is A033581(n). - Kieren MacMillan, Dec 19 2007

For n>1, a(n-1) is the number of aperiodic necklaces (Lyndon words) with k<=3 black beads and n-k white beads. For n=4 we have for example a(3)=3 aperiodic necklaces: BWWW, BBWW and BBBW. BWBW is periodic and is not counted. - Herbert Kociemba, Oct 23 2016

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 9*x^6 + 11*x^7 + 14*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Robert Morris, Minimal percolating sets in bootstrap percolation, arXiv:math/0702370 [math.CO], 2007-2008.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A008747, A033581, A051274, A061347, A071619, A077043.

a:=[1,1,2,3,5,6];; for n in [7..60] do a[n]:=a[n-1]+a[n-2]-a[n-4] -a[n-5]+a[n-6]; od; a; # G. C. Greubel, Aug 03 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^4)/((1-x)*(1-x^2)*(1-x^3)) )); // G. C. Greubel, Aug 03 2019

A008747:=n->ceil((n+1)^2/6): seq(A008747(n), n=0..100); # Wesley Ivan Hurt, Oct 25 2016

CoefficientList[Series[(1+x^4)/((1-x)(1-x^2)(1-x^3)),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,-1,-1,1},{1,1,2,3,5,6},60] (* Harvey P. Dale, Sep 05 2012 *)

Vec((1+x^4)/((1-x)*(1-x^2)*(1-x^3))+O(x^60)) \\ Charles R Greathouse IV, Sep 25 2012

((1+x^4)/((1-x)*(1-x^2)*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019

G.f.: (1+x^4)/((1-x)*(1-x^2)*(1-x^3)).
a(n) = ceiling((n+1)^2/6).
a(n) = (12*n + 23 + 6*n^2 + 9*(-1)^n + 4*A061347(n))/36. - R. J. Mathar, Mar 15 2011
a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=6, a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 5. - Harvey P. Dale, Sep 05 2012
From Michael Somos, Oct 25 2016: (Start)
Euler transform of length 8 sequence [ 1, 1, 1, 1, 0, 0, 0, -1].
a(n) = a(-2-n) for all n in Z.
a(2*n-1) = A071619(n).
a(3*n-1) = 2*A077043(n).
a(n) - a(n-1) = A051274(n). (End)

A129819 Antidiagonal sums of triangular array T: T(j,k) = (k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 7, 8, 12, 14, 19, 21, 27, 30, 37, 40, 48, 52, 61, 65, 75, 80, 91, 96, 108, 114, 127, 133, 147, 154, 169, 176, 192, 200, 217, 225, 243, 252, 271, 280, 300, 310, 331, 341, 363, 374, 397, 408, 432, 444, 469, 481, 507, 520, 547, 560, 588, 602, 631

Offset: 0

Paul Curtz, May 20 2007

nonn

Interleaving of A077043 and A006578.
First differences are in A124072.
If the values of the second, fourth, sixth, ... column are replaced by the corresponding negative values, the antidiagonal sums of the resulting triangular array are 0, 0, 1, 1, -1, -2, -1, -2, -6, -8, -7, -9, ... .
Row sums of triangle A168316 = (1, 1, 3, 4, 7, 8, 12, ...). - Gary W. Adamson, Nov 22 2009

			First seven rows of T are
  0;
  0, 1;
  0, 1, 2;
  0, 1, 3, 2;
  0, 1, 4, 2, 3;
  0, 1, 5, 2, 4, 3;
  0, 1, 6, 2, 5, 3, 4;.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A006578, A077043, A124072, A168316.

m:=59; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=2 to j do if k mod 2 eq 0 then M[j, k]:= k div 2; else M[j, k]:=j-(k div 2); end if; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jul 16 2007

A129819:= func< n | Floor(((n-1)*(3*n+1) +(2*n+5)*((n+1) mod 2))/16) >;
[A129819(n): n in [0..70]]; // G. C. Greubel, Sep 19 2024

CoefficientList[Series[x^2*(1+x^2+x^3)/((1-x)*(1-x^2)*(1-x^4)), {x, 0, 70}], x] (* G. C. Greubel, Sep 19 2024 *)

{vector(59, n, (n-2+n%2)*(n+n%2)/8+floor((n-2-n%2)^2/16))} \\ Klaus Brockhaus, Jul 16 2007

def A129819(n): return ((n-1)*(3*n+1) + (2*n+5)*((n+1)%2))//16
[A129819(n) for n in range(71)] # G. C. Greubel, Sep 19 2024

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n > 6, with a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 3, a(5) = 4, a(6) = 7.
G.f.: x^2*(1+x^2+x^3)/((1-x)^3*(1+x)^2*(1+x^2)).
a(n) = (3/16)*(n+2)*(n+1) - (5/8)*(n+1) + 7/32 + (3/32)*(-1)^n + (1/16)*(n+1)*(-1)^n - (1/8)*cos(n*Pi/2) + (1/8)*sin(n*Pi/2). - Richard Choulet, Nov 27 2008

Edited and extended by Klaus Brockhaus, Jul 16 2007

A109439 Triangle read by rows, in which row n gives coefficients in expansion of ((1 - x^n)/(1 - x))^3.

Original entry on oeis.org

1, 1, 3, 3, 1, 1, 3, 6, 7, 6, 3, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 33, 36, 37, 36, 33, 28, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 36, 42, 46, 48, 48

Offset: 1

Labos Elemer, Jun 30 2005

Sum of n-th row is n^3. The n-th row contains 3n-2 entries. Largest coefficients in rows are listed in A077043. The 255th row describes the distribution of color lattice points in the 765 r+g+b=k planes of the 24-bit RGB-cube with 256^3 points.
Also, the number of cubes of dimension 1 X 1 X 1 needed to build a cube by layers perpendicular to the main diagonal. Each layer is made up of regular triangular numbers T near the summits and truncated T's in the middle. E.g., cube 3^3 is made of layers 1, 3, 6, 7, 6, 3, 1, using T1, T2, T3 and a regularly truncated T4, 7 instead of 10. - M. Dauchez (mdzzdm(AT)yahoo.fr), Aug 31 2005
The n-th row is the third row of the (n+1)-nomial triangle. For example, row 1 (1,3,3,1) is the third row in the binomial triangle; row 5 is the third row of the 6-nomial triangle. - Bob Selcoe, Feb 18 2014
It appears that T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with three different playing cards. - Juan Pablo Herrera P., Nov 04 2016

			Triangle starts:
  1;
  1, 3, 3, 1;
  1, 3, 6, 7, 6, 3, 1;
  1, 3, 6,10,12,12,10, 6, 3, 1;
  1, 3, 6,10,15,18,19,18,15,10, 6, 3, 1;
  1, 3, 6,10,15,21,25,27,27,25,21,15,10, 6, 3, 1;
  1, 3, 6,10,15,21,28,33,36,37,36,33,28,21,15,10, 6, 3, 1.

Cf. A000217, A004737, A045943, A077043.

Flatten[Table[CoefficientList[Series[((1-x^n)/(1-x))^3,{x,1,3*n}],x], {n,1,100}],1]

row(n) = Vec(((1 - x^n)/(1 - x))^3);
tabf(nn) = for (n=1, nn, print(row(n))); \\ Michel Marcus, Oct 12 2016

From Juan Pablo Herrera P., Oct 17 2016: (Start)
T(n,k) = A000217(k+1) = (k+2)!/(k!*2) if 0 <= k < n,
T(n,k) = (9*n-3*n^2+6*k*n-6*k-2*k^2-4)/2 if n-3 < k < 2*n,
T(n,k) = A000217(3n-k-2) = (3*n-k-1)!/((3*(n-1)-k)!*2) if 2*n-3 < k < 3*n-2.
T(n,k) = Sum_{i=k-n+1..k} A004737(T(n,i)),
T(n,k) = Sum_{i=k-n+1..k} (n-|n-i-1|) if n <= k <= 2*n+1. (End)

Offset corrected by Joerg Arndt at the suggestion of Michel Marcus, Oct 12 2016

A102214 Expansion of (1 + 4x + 4x^2)/((1+x)*(1-x)^3).

Original entry on oeis.org

1, 6, 16, 30, 49, 72, 100, 132, 169, 210, 256, 306, 361, 420, 484, 552, 625, 702, 784, 870, 961, 1056, 1156, 1260, 1369, 1482, 1600, 1722, 1849, 1980, 2116, 2256, 2401, 2550, 2704, 2862, 3025, 3192, 3364, 3540, 3721, 3906, 4096, 4290, 4489, 4692, 4900

Offset: 0

Creighton Dement, Feb 17 2005

A floretion-generated sequence.
a(n) gives the number of triples (x,y,x+y) with positive integers satisfying x < y and x + y <= 3*n. - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
Number of different partitions of numbers x + y = z such that {x,y,z} are integers {1,2,3,...,3n} and z > y > x. - Artur Jasinski, Feb 09 2010
Second bisection preceded by zero is A152743. - Bruno Berselli, Oct 25 2011
a(n) has no final digit 3, 7, 8. - Paul Curtz, Mar 04 2020
One odd followed by three evens.
From Paul Curtz, Mar 06 2020: (Start)
b(n) = 0, 1, 6, 16,  30, 49, ... = 0, a(n).
(  25, 12,  4, 0,  1,  6,  16, 30, ...
  -13, -8, -4  1,  5, 10,  14, 19, ...
    5,  4,  5, 4,  5,  4,   5,  4, ... .)
b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .
A154589(n) are in the main diagonal of b(n) and b(-n). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A016778, A038764, A001859, A006578, A069905, A077043, A274221, A330707.

Cf. A000326, A016789, A152743, A001651, A032766, A047237, A047251, A154589.

[(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011

aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y + 1, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 20}]; aa (* Artur Jasinski, Feb 09 2010 *)

a(n)=(6*n*(3*n+4)+(-1)^n+7)/8 \\ Charles R Greathouse IV, Apr 16 2020

G.f.: -(4*x^2 + 4*x + 1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).
a(2n) = A016778(n) = (3n+1)^2.
a(n) + a(n+1) = A038764(n+1).
a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
a(n) = (6*n*(3*n+4) + (-1)^n+7)/8. - Bruno Berselli, Oct 25 2011
a(n) = A198392(n) + A198392(n-1). - Bruno Berselli, Nov 06 2011
From Paul Curtz, Mar 04 2020: (Start)
a(n) = A006578(n) + A001859(n) + A077043(n+1).
a(n) = A274221(2+2*n).
a(20+n) - a(n) = 30*(32+3*n).
a(1+2*n) = 3*(1+n)*(2+3*n).
a(n) = A047237(n) * A047251(n).
a(n) = A001651(n+1) * A032766(n).(End)
E.g.f.: ((4 + 21*x + 9*x^2)*cosh(x) + 3*(1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020

Definition rewritten by Bruno Berselli, Oct 25 2011

A220098 Manhattan distances between 2n and 1 in the double spiral with positive integers and 1 at the center.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 13

Offset: 1

Alex Ratushnyak, Dec 04 2012

Double spiral begins:
.
  82---84---86---88---90---92---94---96---98
   |
  80   51---53---55---57---59---61---63---65
   |    |                                  |
  78   49   26---28---30---32---34---36   67
   |    |    |                        |    |
  76   47   24   11---13---15---17   38   69
   |    |    |    |              |    |    |
  74   45   22    9    2----4   19   40   71
   |    |    |    |    |    |    |    |    |
  72   43   20    7    1    6   21   42   73
   |    |    |    |    |    |    |    |    |
  70   41   18    5----3    8   23   44   75
   |    |    |              |    |    |    |
  68   39   16---14---12---10   25   46   77
   |    |                        |    |    |
  66   37---35---33---31---29---27   48   79
   |                                  |    |
  64---62---60---58---56---54---52---50   81
                                           |
  99---97---95---93---91---89---87---85---83

			From _Philippe Deléham_, Mar 08 2013: (Start)
As a square array, this begins:
  1,  1,  2,  2,  3,  3,  4,  4,  5, ...
  2,  3,  3,  4,  4,  5,  5,  6,  6, ...
  2,  4,  5,  5,  6,  6,  7,  7,  8, ...
  3,  4,  6,  7,  7,  8,  8,  9,  9, ...
  3,  5,  6,  8,  9,  9, 10, 10, 11, ...
  4,  5,  7,  8, 10, 11, 11, 12, 12, ...
  4,  6,  7,  9, 10, 12, 13, 13, 14, ...
  5,  6,  8,  9, 11, 12, 14, 15, 15, ..., etc.
As a triangle, this begins:
  1
  2, 1
  2, 3, 2
  3, 4, 3, 2
  3, 4, 5, 4, 3
  4, 5, 6, 5, 4, 3, etc. (End)

Cf. A053615, A077043, A128217, A214526.

#include 
#define SIZE 20
int grid[SIZE][SIZE];
int direction[] = {0, -1,  1, 0, 0, 1, -1, 0};
main() {
  int i, j, x1, y1, x2, y2, stepSize;
  int direction1pos=0, direction2pos=4, val;
  x1 = y1 = x2 = y2 = SIZE/2;
  for (val=grid[y1][x1]=1, stepSize=0; ; ++stepSize) {
    if (x1<1 || x1>=SIZE-1 || x2<1 || x2>=SIZE-1) break;
    if (y1<1 || y1>=SIZE-1 || y2<1 || y2>=SIZE-1) break;
    for (i=stepSize|1; i; ++val,--i) {
      x1 += direction[direction1pos  ];
      y1 += direction[direction1pos+1];
      x2 += direction[direction2pos  ];
      y2 += direction[direction2pos+1];
      grid[y1][x1] = val*2;
      grid[y2][x2] = val*2+1;
      printf("%d, ",abs(x1-SIZE/2)+abs(y1-SIZE/2));
    }
    direction1pos = (direction1pos+2) & 7;
    direction2pos = (direction2pos+2) & 7;
  }
  for (i=0; i

step(v, m) = concat(v, vector(m, k, 1+v[#v-k+1]))
a(max_n) = {my(v=[0], k=1); while(#v < max_n+1, v=step(v,k); k++); v[2..max_n+1]} \\ Thomas Scheuerle, Jan 07 2025

A053615(n) = if(n<1, 0, sqrtint(n) - A053615(n - sqrtint(n)))
a(n) = A053615(floor( floor( (sqrtint(n*8) + 1)/2 )^2/2 ) + n) \\ Thomas Scheuerle, Jan 07 2025

abs( a(n) - a(n-1) ) = 1.
From Thomas Scheuerle, Jan 07 2025: (Start)
a(n*(n+1)/2 - k) = 1 + a(n*(n-1)/2 + k) with a(0) = 0 and for 0 <= k < n.
a(n) = A053615(A128217(n+1)). (End)

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)).

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A274581 Number T(n,k) of set partitions of [n] with alternating parity of elements and exactly k blocks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A132894 Number of (1,0) steps in all paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., in all length-n left factors of Motzkin paths).

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A235988 Sum of the partition parts of 3n into 3 parts.

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A008747 Expansion of (1+x^4)/((1-x)*(1-x^2)*(1-x^3)).

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A129819 Antidiagonal sums of triangular array T: T(j,k) = (k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.

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A008747 Expansion of (1+x^4)/((1-x)(1-x^2)(1-x^3)).

A102214 Expansion of (1 + 4x + 4x^2)/((1+x)*(1-x)^3).