A097613 -id:A097613 - cp's OEIS frontend

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.

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

A122919 Inverse of Riordan array (1/(1+x+x^2),x/(1+x)^2).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 9, 10, 5, 1, 28, 34, 21, 7, 1, 90, 117, 83, 36, 9, 1, 297, 407, 319, 164, 55, 11, 1, 1001, 1430, 1209, 702, 285, 78, 13, 1, 3432, 5070, 4550, 2898, 1350, 454, 105, 15, 1, 11934, 18122, 17068, 11696, 6052

Offset: 0

Paul Barry, Sep 19 2006

Row sums are A097613. Diagonal sums are A122920. Inverse of A122917.

			Triangle begins
1,
1, 1,
3, 3, 1,
9, 10, 5, 1,
28, 34, 21, 7, 1,
90, 117, 83, 36, 9, 1,
297, 407, 319, 164, 55, 11, 1,
1001, 1430, 1209, 702, 285, 78, 13, 1

Riordan array ((1-3x+2x^2-(1-x)sqrt(1-4x))/(2x^2),(1-2x-sqrt(1-4x))/(2x))

A225015 Number of sawtooth patterns of length 1 in all Dyck paths of semilength n.

Original entry on oeis.org

0, 1, 1, 5, 18, 66, 245, 918, 3465, 13156, 50193, 192270, 739024, 2848860, 11009778, 42642460, 165480975, 643281480, 2504501625, 9764299710, 38115568260, 148955040300, 582714871830, 2281745337300, 8942420595810, 35074414899576, 137672461877850, 540756483094828

Offset: 0

David Scambler, Apr 23 2013

nonn

A sawtooth pattern of length 1 is UD not followed by UD.

First differences of A024482.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A024482, A097613.

A024482:= func< n | (3*n-2)*Catalan(n-1)/2 >;
A225015:= func< n | n le 2 select Floor((n+1)/2) else A024482(n) - A024482(n-1) >;
[A225015(n): n in [0..40]]; // G. C. Greubel, Apr 03 2024

a:= proc(n) option remember; `if`(n<4, [0, 1, 1, 5][n+1],
       (n-1)*(3*n-4)*(4*n-10)*a(n-1)/(n*(n-2)*(3*n-7)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 24 2013

Join[{0, 0, 1}, Table[(Binomial[2n, n]-Binomial[2n-2, n-1])/2, {n, 2, 25}]] // Differences (* Jean-François Alcover, Nov 12 2020 *)

def A024482(n): return (3*n-2)*catalan_number(n-1)/2
def A225015(n): return floor((n+1)/2) if n<3 else A024482(n) - A024482(n-1)
[A225015(n) for n in range(41)] # G. C. Greubel, Apr 03 2024

a(0)=0, a(1)=1, a(n) = A024482(n) - A024482(n-1) for n >= 2.
From G. C. Greubel, Apr 03 2024: (Start)
G.f.: (1-x)^2*(1 - sqrt(1-4*x))/(2*sqrt(1-4*x)).
E.g.f.: -(1/4)*(2-4*x+x^2) + (1/12)*Exp(2*x)*((6-12*x+43*x^2-24*x^3) *BesselI(0, 2*x) - 4*x*(7-5*x)*BesselI(1,2*x) - 3*x^2*(13-8*x)* BesselI(2,2*x)). (End)

A108746 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having k peaks that are not of the form uudd (here u=(1,1), d=(1,-1)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 2, 4, 4, 3, 1, 4, 9, 13, 9, 6, 1, 8, 23, 34, 35, 21, 10, 1, 17, 56, 97, 111, 86, 46, 15, 1, 37, 138, 272, 347, 321, 201, 92, 21, 1, 82, 344, 749, 1083, 1111, 846, 449, 169, 28, 1, 185, 859, 2063, 3289, 3786, 3255, 2080, 953, 289, 36, 1, 423, 2154

Offset: 0

Emeric Deutsch, Jun 23 2005

Also number of ordered trees with n edges and having k leaves that are endpoints of branches of length 1 (i.e. leaf is child of the root or of a branchnode). Row sums are the Catalan numbers (A000108). Column 0 yields A004148. Sum(kT(n,k),k=0..n)=A097613(n).

			T(4,2)=4 because we have uu(ud)(ud)dd, uudd(ud)(ud), (ud)uudd(ud) and
(ud)(ud)uudd (the peaks that are not of the form uudd are shown between parentheses).
Triangle begins:
1;
0,1;
1,0,1;
1,2,1,1;
2,4,4,3,1;
4,9,13,9,6,1

Cf. A000108, A004148, A097613.

G:=1/2/z*(1-z^2+z-t*z+z^2*t-sqrt(1-z^2+z^4-2*z^3+4*z^3*t-2*z^4*t+t^2*z^2-2*t^2*z^3+z^4*t^2-2*t*z-2*z)): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

G.f.=G=G(t, z) satisfies G=1+zG(G-1+t-tz+z).

cp's OEIS Frontend

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.

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

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A122919 Inverse of Riordan array (1/(1+x+x^2),x/(1+x)^2).

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A225015 Number of sawtooth patterns of length 1 in all Dyck paths of semilength n.

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A108746 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having k peaks that are not of the form uudd (here u=(1,1), d=(1,-1)).

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Maple

Formula

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