A051924 -id:A051924 - cp's OEIS frontend

A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870

Offset: 0

Peter Luschny, Aug 05 2009

Triangle read by rows. For n >= 0, k >= 0 let T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040).

This is also the square array of central binomial coefficients A000984 in column 0 and higher (first: A051924, second, etc.) differences in subsequent columns, read by antidiagonals. - M. F. Hasler, Nov 15 2019

			Triangle begins
    1;
    1,   2;
    3,   4,   6;
    7,  10,  14,  20;
   19,  26,  36,  50,  70;
   51,  70,  96, 132, 182, 252;
  141, 192, 262, 358, 490, 672, 924;
From _M. F. Hasler_, Nov 15 2019: (Start)
The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:
     1   1    3    7    19    51 ...
     2   4   10   26    70   192 ...
     6  14   36   96   262   726 ...
    20  50  132  358   988  2760 ...
    70 182  490 1346  3748 10540 ...
   252 672 1836 5094 14288 40404 ...
  (...)
Read by falling antidiagonals this yields the same sequence. (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Row sums are A163774. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

A265283 Number of ON (black) cells in the n-th iteration of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68

Offset: 0

Robert Price, Dec 06 2015

From Gus Wiseman, Apr 13 2019: (Start)
Also the number of integer partitions of n + 3 such that lesser of the maximum part and the number of parts is 2. The Heinz numbers of these partitions are given by A325229. For example, the a(0) = 1 through a(7) = 10 partitions are:
  (21)  (22)   (32)    (33)     (43)      (44)       (54)        (55)
        (31)   (41)    (42)     (52)      (53)       (63)        (64)
        (211)  (221)   (51)     (61)      (62)       (72)        (73)
               (2111)  (222)    (2221)    (71)       (81)        (82)
                       (2211)   (22111)   (2222)     (22221)     (91)
                       (21111)  (211111)  (22211)    (222111)    (22222)
                                          (221111)   (2211111)   (222211)
                                          (2111111)  (21111111)  (2221111)
                                                                 (22111111)
                                                                 (211111111)
(End)

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:
                        1                          =  1
                      1 1 1                        =  3
                    1 1 . 1 1                      =  4
                  1 1 1 . 1 1 1                    =  6
                1 1 . 1 . 1 . 1 1                  =  6
              1 1 1 . 1 . 1 . 1 1 1                =  8
            1 1 . 1 . 1 . 1 . 1 . 1 1              =  8
          1 1 1 . 1 . 1 . 1 . 1 . 1 1 1            = 10
        1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1          = 10
      1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1        = 12
    1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1      = 12
  1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1    = 14
1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1  = 14
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Column k = 2 of A325227.

Cf. A004526, A051924, A118102, A263297, A325193, A325225, A325228, A325229, A325232.

rule = 94; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1},0},rows-1,{All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]
Total /@ CellularAutomaton[94, {{1}, 0}, 65] (* Michael De Vlieger, Dec 14 2015 *)

Conjectures from Colin Barker, Dec 07 2015 and Apr 16 2019: (Start)
a(n) = (5-(-1)^n+2*n)/2 = A213222(n+3) for n>1.
a(n) = n+2 for n>1 and even.
a(n) = n+3 for n>1 and odd.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>2.
G.f.: (1+2*x-x^4) / ((1-x)^2*(1+x)).
(End)

A265612 a(n) = CatalanNumber(n+1)n(1+3n)/(6+2n).

Original entry on oeis.org

0, 1, 7, 35, 156, 660, 2717, 11011, 44200, 176358, 700910, 2778446, 10994920, 43459650, 171655785, 677688675, 2674776720, 10555815270, 41656918050, 164401379610, 648887951400, 2561511781920, 10113397410402, 39937416869070, 157743149913776, 623178050662300

Offset: 0

Peter Luschny, Dec 15 2015

nonn

This is row n=7 in the array A(n,k) = (rf(k+n-2,k-1)-(k-1)*(k-2)*rf(k+n-2, k-3))/ (k-1)! if n>=3 and A(n,0)=0, A(n,1)=1, A(n,2)=n; rf(n,k) denotes the rising factorial. See the cross-references for other values of n and the table in A264357.

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

Cf. A000108, A007946, A097613, A051960, A029651, A051924, A129869, A220101, A264357, A265613.

A265612 := n -> 2*4^n*GAMMA(3/2+n)*n*(1+3*n)/(sqrt(Pi)*GAMMA(4+n)):
seq(simplify(A265612(n)), n=0..25);

Table[SeriesCoefficient[(5 x + (I (x - 1) (7 x - 2))/Sqrt[4 x - 1] - 2 - x^2)/(2 x^3), {x, 0, n}], {n, 0, 25}] (* or *)
Table[2*4^n Gamma[3/2 + n] n (1 + 3 n)/(Sqrt[Pi] Gamma[4 + n]), {n, 0, 25}] (* or *)
Table[CatalanNumber[n + 1] n ((1 + 3 n)/(6 + 2 n)), {n, 0, 25}] (* Michael De Vlieger, Dec 15 2015 *)

for(n=0,25, print1(round(2*4^n*gamma(3/2+n)*n*(1+3*n)/(sqrt(Pi)*gamma(4+n))), ", ")) \\ G. C. Greubel, Feb 06 2017

a = lambda n: catalan_number(n+1)*n*(1+3*n)/(6+2*n)
[a(n) for n in range(26)]

G.f.: (5*x+(I*(x-1)*(7*x-2))/sqrt(4*x-1)-2-x^2)/(2*x^3).
a(n) = 2*4^n*Gamma(3/2+n)*n*(1+3*n)/(sqrt(Pi)*Gamma(4+n)).
a(n) = (rf(5+n, n-1)-(n-1)*(n-2)*rf(5+n, n-3))/(n-1)! for n>=3, rf(n,k) the rising factorial.
a(n) = a(n-1)*(2*n*(1+3*n)*(1+2*n)/((n-1)*(3*n-2)*(3+n))) for n>=2.
a(n) ~  4^n*(6-(127/4)/n+(7995/64)/n^2-(223405/512)/n^3+(23501457/16384)/n^4-...) /sqrt(n*Pi).
a(n) = [x^n] x*(1 + x)/(1 - x)^(n+4). - Ilya Gutkovskiy, Oct 09 2017

A051944 a(n) = C(n)(4n+1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 5, 18, 65, 238, 882, 3300, 12441, 47190, 179894, 688636, 2645370, 10192588, 39373700, 152443080, 591385545, 2298248550, 8945490510, 34867625100, 136079265630, 531693754020, 2079632696700, 8141948163960, 31904544069450, 125120702290428, 491056586546652

Offset: 0

Barry E. Williams, Dec 20 1999

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=4 of A330965.

Cf. A016777, A000108, A051924.

[Catalan(n)*(4*n+1):n in [0..30] ]; // Marius A. Burtea, Jan 05 2020

R:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (3 - 4*x - 3*Sqrt(1 - 4*x))/(2*x*Sqrt(1 - 4*x)))) ); // Marius A. Burtea, Jan 05 2020

Table[CatalanNumber[n](4n+1),{n,0,30}] (* Harvey P. Dale, Feb 21 2022 *)

{a(n)=if(n<0, 0, (4*n+1)*binomial(2*n,n)/(n+1))} /* Michael Somos, Sep 17 2006 */

The Hankel determinant transform is A025172(n-1). - Michael Somos, Sep 17 2006
-(n+1)*(4*n-3)*a(n) + 2*(4*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 19 2014
G.f.: (3 - 4*x - 3*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 3*binomial(2*n, n-1) = A000984(n) + 3*A001791(n).
a(n) ~ 4^(n+1)/sqrt(Pi*n). (End)

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2020

A080721 Triangle of binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1)).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 9, 21, 14, 1, 16, 66, 100, 50, 1, 25, 160, 410, 455, 182, 1, 36, 330, 1260, 2310, 2016, 672, 1, 49, 609, 3220, 8610, 12222, 8778, 2508, 1, 64, 1036, 7224, 26250, 53592, 61908, 37752, 9438, 1, 81, 1656, 14700, 69300, 189882, 312312, 303732, 160875, 35750, 1

Offset: 0

Paul Boddington, Mar 07 2003

For n>1 and 0 <= k <= n, a(n,k) is the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's 'cluster algebra' of finite type D_n.

Triangle of f-vectors of the simplicial complexes dual to the generalized associahedra of type D_n (n >= 2). See A145903 for the corresponding triangle of h-vectors. For the triangles of f-vectors of type A and type B associahedra see A033282 and A063007 respectively. [Peter Bala, Oct 28 2008]

			Contribution from _Peter Bala_, Oct 28 2008: (Start)
Triangle begins
n\k|..0....1....2....3....4....5
================================
0..|..1
1..|..1....1
2..|..1....4....4
3..|..1....9...21...14
4..|..1...16...66..100...50
5..|..1...25..160..410..455..182
...
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, arXiv:math/0104151 [math.RT], 2001; J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008. [From _Peter Bala_, Oct 28 2008]
Yasuaki Gyoda, Positive cluster complexes and tau-tilting simplicial complexes of cluster-tilted algebras of finite type, arXiv:2105.07974 [math.RT], 2021, see page 34.

Cf. A033282, A063007.

A051924 (main diagonal), A145903( h-vectors type D associahedra). [From Peter Bala, Oct 28 2008]

A080721 := proc(n,k)
    binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1))
end proc: # R. J. Mathar, Mar 22 2013

Flatten[Table[Binomial[n,k](Binomial[n+k,k]-Binomial[Abs[n+k-2],k-1]),{n,0,10},{k,0,n}]] (* Harvey P. Dale, Feb 20 2013 *)

T(n,k)=binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1))
for (n=0, 10, for (k=0,n, print1(T(n,k),", ")));
/* Joerg Arndt, Feb 21 2013 */

A145903 Generalized Narayana numbers for root systems of type D_n. Triangle of h-vectors of type D associahedra.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 24, 12, 1, 1, 20, 70, 70, 20, 1, 1, 30, 165, 280, 165, 30, 1, 1, 42, 336, 875, 875, 336, 42, 1, 1, 56, 616, 2296, 3500, 2296, 616, 56, 1, 1, 72, 1044, 5292, 11466, 11466, 5292, 1044, 72, 1

Offset: 0

Peter Bala, Oct 28 2008

The generalized Narayana numbers of type D_n (row n of this triangle) are defined as the entries of the h-vector of the simplicial complex dual to the generalized associahedron of type D_n [Fomin & Reading, p.60]. For the corresponding triangle of f-vectors see A080721. For Narayana numbers of root systems of type A and type B see A001263 and A008459 respectively.

			Root systems of type D_n are defined only for n >= 2. It seems convenient to complete the array to form a lower unit triangular matrix.
Triangle starts
n\k|..0....1....2....3....4....5....6
=====================================
0..|..1
1..|..1....1
2..|..1....2....1
3..|..1....6....6....1
4..|..1...12...24...12....1
5..|..1...20...70...70...20....1
6..|..1...30..165..280..165...30....1
...

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 12.

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005-2008.

A001263, A008459, A051924 (row sums), A080721.

For n >= 2, T(n,k) = binomial(n,k)^2 - n/(n-1)*binomial(n-1,k-1)*binomial(n-1,k).

A264357 Array A(r, n) of number of independent components of a symmetric traceless tensor of rank r and dimension n, written as triangle T(n, r) = A(r, n-r+2), n >= 1, r = 2..n+1.

Original entry on oeis.org

0, 2, 0, 5, 2, 0, 9, 7, 2, 0, 14, 16, 9, 2, 0, 20, 30, 25, 11, 2, 0, 27, 50, 55, 36, 13, 2, 0, 35, 77, 105, 91, 49, 15, 2, 0, 44, 112, 182, 196, 140, 64, 17, 2, 0, 54, 156, 294, 378, 336, 204, 81, 19, 2, 0

Offset: 1

Wolfdieter Lang, Dec 10 2015

A (totally) symmetric traceless tensor of rank r >= 2 and dimension n >= 1 is irreducible.
The array of the number of independent components of a rank r symmetric traceless tensor A(r, n), for r >= 2 and n >=1, is given by risefac(n,r)/r! - risefac(n,r-2)/(r-2)!, where the first term gives the number of independent components of a symmetric tensors of rank r (see a Dec 10 2015 comment under A135278) and the second term is the number of constraints from the tracelessness requirement. The tensor has to be traceless in each pair of indices.
The first rows of the array A, or the first columns (without the first r-2 zeros) of the triangle T are for r = 2..6: A000096, A005581, A005582, A005583, A005584.
Equals A115241 with the first column of positive integers removed. - Georg Fischer, Jul 26 2023

			The array A(r, n) starts:
   r\n 1 2  3   4   5    6    7     8     9    10 ...
   2:  0 2  5   9  14   20   27    35    44    54
   3:  0 2  7  16  30   50   77   112   156   210
   4:  0 2  9  25  55  105  182   294   450   660
   5:  0 2 11  36  91  196  378   672  1122  1782
   6:  0 2 13  49 140  336  714  1386  2508  4290
   7:  0 2 15  64 204  540 1254  2640  5148  9438
   8:  0 2 17  81 285  825 2079  4719  9867 19305
   9:  0 2 19 100 385 1210 3289  8008 17875 37180
  10:  0 2 21 121 506 1716 5005 13013 30888 68068
  ...
The triangle T(n, r) starts:
   n\r  2   3   4   5   6   7  8  9 10 11 ...
   1:   0
   2:   2   0
   3:   5   2   0
   4:   9   7   2   0
   5:  14  16   9   2   0
   6:  20  30  25  11   2   0
   7:  27  50  55  36  13   2  0
   8:  35  77 105  91  49  15  2  0
   9:  44 112 182 196 140  64 17  2  0
  10:  54 156 294 378 336 204 81 19  2  0
  ...
A(r, 1) = 0 , r >= 2, because a symmetric rank r tensor t of dimension one has one component t(1,1,...,1) (r 1's) and if the traces vanish then t vanishes.
A(3, 2) = 2 because a symmetric rank 3 tensor t with three indices taking values from 1 or 2 (n=2) has the four independent components t(1,1,1), t(1,1,2), t(1,2,2), t(2,2,2), and (invoking symmetry) the vanishing traces are Sum_{j=1..2} t(j,j,1) = 0 and Sum_{j=1..2} t(j,j,2) = 0. These are two constraints, which can be used to eliminate, say, t(1,1,1) and t(2,2,2), leaving 2 = A(3, 2) independent components, say, t(1,1,2) and t(1,2,2).
From _Peter Luschny_, Dec 14 2015: (Start)
The diagonals diag(n, k) start:
   k\n  0       1       2       3       4      5       6
   0:   0,      2,      9,     36,    140,   540,   2079, ... A007946
   1:   2,      7,     25,     91,    336,  1254,   4719, ... A097613
   2:   5,     16,     55,    196,    714,  2640,   9867, ... A051960
   3:   9,     30,    105,    378,   1386,  5148,  19305, ... A029651
   4:  14,     50,    182,    672,   2508,  9438,  35750, ... A051924
   5:  20,     77,    294,   1122,   4290, 16445,  63206, ... A129869
   6:  27,    112,    450,   1782,   7007, 27456, 107406, ... A220101
   7:  35,    156,    660,   2717,  11011, 44200, 176358, ... A265612
   8:  44,    210,    935,    4004, 16744, 68952, 281010, ... A265613
  A000096,A005581,A005582,A005583,A005584.
(End)

Cf. A000096, A005581, A005582, A005583, A005584, A007946, A024482, A029651, A051924, A051960, A097613, A115241, A129869, A135278, A220101, A265612, A265613.

A[r_, n_] := Pochhammer[n, r]/r! - Pochhammer[n, r-2]/(r-2)!;
T[n_, r_] := A[r, n-r+2];
Table[T[n, r], {n, 1, 10}, {r, 2, n+1}] (* Jean-François Alcover, Jun 28 2019 *)

A = lambda r, n: rising_factorial(n,r)/factorial(r) - rising_factorial(n,r-2)/factorial(r-2)
for r in (2..10): [A(r,n) for n in (1..10)] # Peter Luschny, Dec 13 2015

T(n, r) = A(r, n-r+2) with the array A(r, n) = risefac(n,r)/r! - risefac(n,r-2)/(r-2)! where the rising factorial risefac(n,k) = Product_{j=0..k-1} (n+j) and risefac(n,0) = 1.
From Peter Luschny, Dec 14 2015: (Start)
A(n+2, n+1) = A007946(n-1) = CatalanNumber(n)*3*n*(n+1)/(n+2) for n>=0.
A(n+2, n+2) = A024482(n+2) = A097613(n+2) = CatalanNumber(n+1)*(3*n+4)/2 for n>=0.
A(n+2, n+3) = A051960(n+1) = CatalanNumber(n+1)*(3*n+5) for n>=0.
A(n+2, n+4) = A029651(n+2) = CatalanNumber(n+1)*(6*n+9) for n>=0.
A(n+2, n+5) = A051924(n+3) = CatalanNumber(n+2)*(3*n+7) for n>=0.
A(n+2, n+6) = A129869(n+4) = CatalanNumber(n+2)*(3*n+8)*(2*n+5)/(n+4) for n>=0.
A(n+2, n+7) = A220101(n+4) = CatalanNumber(n+3)*(3*(n+3)^2)/(n+5) for n>=0.
A(n+2, n+8) = CatalanNumber(n+4)*(n+3)*(3*n+10)/(2*n+12) for n>=0.
Let for n>=0 and k>=0 diag(n,k) = A(k+2,n+k+1) and G(n,k) = 2^(k+2*n)*Gamma((3-(-1)^k+2*k+4*n)/4)/(sqrt(Pi)*Gamma(k+n+0^k)) then
diag(n,0) = G(n,0)*(n*3)/(n+2),
diag(n,1) = G(n,1)*(3*n+4)/((n+1)*(n+2)),
diag(n,2) = G(n,2)*(3*n+5)/(n+2),
diag(n,3) = G(n,3)*3,
diag(n,4) = G(n,4)*(3*n+7),
diag(n,5) = G(n,5)*(3*n+8),
diag(n,6) = G(n,6)*3*(3+n)^2,
diag(n,7) = G(n,7)*(3+n)*(10+3*n). (End)

A265613 a(n) = CatalanNumber(n+1)n(3n^2+5n+2)/((4+n)*(3+n)).

Original entry on oeis.org

0, 1, 8, 44, 210, 935, 4004, 16744, 68952, 281010, 1136960, 4576264, 18349630, 73370115, 292746300, 1166182800, 4639918800, 18443677230, 73261092240, 290845019400, 1154169552900, 4578702310182, 18159992594568, 72014135814704, 285542883894800, 1132125641947300

Offset: 0

Peter Luschny, Dec 15 2015

nonn

This is row n=8 in the array A(n,k) = (rf(k+n-2,k-1)-(k-1)*(k-2)*rf(k+n-2, k-3))/ (k-1)! if n>=3 and A(n,0)=0, A(n,1)=1, A(n,2)=n; rf(n,k) denotes the rising factorial. See the cross-references for other values of n and the table in A264357.

Robert Israel, Table of n, a(n) for n = 0..1630

Cf. A000108, A007946, A097613, A051960, A029651, A051924, A129869, A220101, A264357, A265612.

A265613 := n -> (4*4^n*n*(n+1)*(3*n+2)*GAMMA(n+3/2))/(sqrt(Pi)*GAMMA(n+5)):
seq(simplify(A265613(n)), n=0..25);

Table[SeriesCoefficient[I (14 x^2 + I Sqrt[4 x - 1] (4 x^2 - 7 x + 2) - 11 x + 2 (1 - x^3))/(2 x^4 Sqrt[4 x - 1]), {x, 0, n}], {n, 0, 25}]
(* or *)
Table[(4^(n + 1) n (n + 1) (3 n + 2) Gamma[n + 3/2])/(Sqrt[Pi] Gamma[n + 5]), {n, 0, 25}] (* or *)
Table[CatalanNumber(n+1) n (3 n^2 + 5 n + 2)/((4 + n) (3 + n)), {n, 0, 25}] (* Michael De Vlieger, Dec 15 2015 *)

a = lambda n: catalan_number(n+1)*n*(3*n^2+5*n+2)/((4+n)*(3+n))
[a(n) for n in range(26)]

G.f.: I*(14*x^2+I*sqrt(4*x-1)*(4*x^2-7*x+2)-11*x+2*(1-x^3))/(2*x^4*sqrt(4*x-1)).
a(n) = (4^(n+1)*n*(n+1)*(3*n+2)*Gamma(n+3/2))/(sqrt(Pi)*Gamma(n+5)).
a(n) = (rf(n+6, n-1)-(n-1)*(n-2)*rf(n+6, n-3))/(n-1)! for n>=3, rf(n,k) the rising factorial.
a(n) = a(n-1)*((2*(n+1))*(3*n+2)*(1+2*n)/((n-1)*(3*n-1)*(4+n))) for n>=2.
a(n) ~ 4^n*(12-(191/2)/n+(17595/32)/n^2-(705005/256)/n^3+(104705937/8192)/ n^4-...)/sqrt(n*Pi).
a(n) = [x^n] x*(1 + x)/(1 - x)^(n+5). - Ilya Gutkovskiy, Oct 09 2017

A325194 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with co-rank n - k, where co-rank is the greater of the length and the largest part.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 13 2019

			Triangle begins:
  1
  0  0
  0  1  0
  0  0  0  0
  0  0  2  0  0
  0  0  0  1  0  0
  0  0  0  2  1  0  0
  0  0  0  0  2  0  0  0
  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  4  2  0  0  0
  0  0  0  0  0  0  2  5  1  0  0  0
  0  0  0  0  0  0  2  4  7  1  0  0  0
  0  0  0  0  0  0  0  2  6  6  0  0  0  0
  0  0  0  0  0  0  0  2  4  9  7  0  0  0  0
  0  0  0  0  0  0  0  0  2  6 11  5  0  0  0  0
  0  0  0  0  0  0  0  0  2  4 10 14  5  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  6 13 15  3  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  4 10 19 17  2  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  6 14 22 17  1  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  4 10 21 29 17  1  0  0  0  0
Row n = 16 counts the following partitions:
  (8)         (72)       (64)      (533)    (444)
  (11111111)  (711)      (622)     (542)    (3333)
              (2211111)  (631)     (551)    (4332)
              (3111111)  (6211)    (5222)   (4422)
                         (61111)   (5321)   (4431)
                         (222211)  (5411)
                         (322111)  (32222)
                         (331111)  (33221)
                         (421111)  (33311)
                         (511111)  (42221)
                                   (43211)
                                   (44111)
                                   (52211)
                                   (53111)

Column sums are A000041. Row sums are A325193.

Cf. A051924, A071724, A096771, A115720, A263297, A325178, A325192.

Table[Length[Select[IntegerPartitions[k],Max[Length[#],Max[#]]==n-k&]],{n,0,10},{k,0,n}]

A325223 Sum of the prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

Also the number of squares in the Young diagram of the integer partition with Heinz number n after the first row or the first column, whichever is larger, is removed. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			88 has 4 prime indices {1,1,1,5} with sum 8 and maximum 5, so a(88) = 8 - max(4,5) = 3.

FindStat, St000784: The maximum of the length and the largest part of the integer partition

Positions of 0's are A174090. Positions of 1's are A325231.

Cf. A001222, A051924, A052126, A056239, A061395, A064989, A065770, A112798, A257990, A263297, A325134, A325169, A325224, A325225, A325227.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[primeMS[n]]-Max[Length[primeMS[n]],Max[primeMS[n]]],{n,100}]

a(n) = A056239(n) - max(A001222(n), A061395(n)) = A056239(n) - A263297(n).

cp's OEIS Frontend

A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A265283 Number of ON (black) cells in the n-th iteration of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A265612 a(n) = CatalanNumber(n+1)*n*(1+3*n)/(6+2*n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A051944 a(n) = C(n)*(4*n+1) where C(n) = Catalan numbers (A000108).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula

Extensions

A080721 Triangle of binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A145903 Generalized Narayana numbers for root systems of type D_n. Triangle of h-vectors of type D associahedra.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A264357 Array A(r, n) of number of independent components of a symmetric traceless tensor of rank r and dimension n, written as triangle T(n, r) = A(r, n-r+2), n >= 1, r = 2..n+1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A265612 a(n) = CatalanNumber(n+1)n(1+3n)/(6+2n).

A051944 a(n) = C(n)(4n+1) where C(n) = Catalan numbers (A000108).

A265613 a(n) = CatalanNumber(n+1)n(3n^2+5n+2)/((4+n)*(3+n)).