A126120 -id:A126120 - cp's OEIS frontend

A138356 Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(4).

1, 1, 2, 4, 10, 27, 82, 268, 940, 3476, 13448, 53968, 223412, 949535, 4128594, 18310972, 82645012, 378851428, 1760998280, 8288679056, 39457907128, 189784872428, 921472827272, 4512940614960, 22279014978544, 110797225212112

Offset: 0

Andrew V. Sutherland, Mar 17 2008

nonn

Let the random variable X be the coefficient of t^2 in the characteristic polynomial det(tI-A) of a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic). Then a(n) = E[X^n].
Let L_p(T) be the L-polynomial (numerator of the zeta function) of a genus 2 curve C. Under a generalized Sato-Tate conjecture, for almost all C,
a(n) is the n-th moment of the coefficient of t^2 in L_p(t/sqrt(p)), as p varies.
See A095922 for central moments.

			a(3) = 4 because E[X^3] = 4 for X the t^2 coeff of det(tI-A) in USp(4).
a(3) = 1*2^3*(1*1-0^2) + 3*2^2*(0*0-1^2) + 3*2^1*(1*2-0^2) + 1*2^0*(0*0-2^2) = 8 - 12 + 12 - 4 = 4.

Kiran S. Kedlaya, Andrew V. Sutherland, Computing L-series of hyperelliptic curves, arXiv:0801.2778 [math.NT], 2008-2012; Algorithmic Number Theory Symposium--ANTS VIII, 2008.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Nicholas M. Katz and Peter Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, AMS, 1999.

Cf. A095922, A138349.

a(n) = (1/2)Integral_{x=0..Pi,y=0..Pi}(4cos(x)cos(y)+2)^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy.

a(n) = Sum_{i=0..n}binomial(n,i)2^{n-i}*(A126120(i)*A126120(i+2)-A126120(i+1)^2).

A302182 Number of 3D walks of type abc.

Original entry on oeis.org

1, 1, 5, 12, 62, 200, 1065, 3990, 21714, 89082, 492366, 2147376, 12004740, 54718092, 308559537, 1454116950, 8255788970, 39935276810, 227976044010, 1126178350440, 6457854821340, 32456552441040, 186814834574550, 952569927106980, 5500292590186380, 28391993275117500

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A018224, A126120, A135394, A302181.

from math import comb as binomial
def row(n: int) -> list[int]:
    return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(n-k, (n-k)//2)**2 for k in range(n+1))
for n in range(26): print(row(n)) # Mélika Tebni, Nov 27 2024

From Mélika Tebni, Nov 27 2024: (Start)
a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A018224(n-k).
a(2*n+1) = A135394(n) / (2*n+2).
a(2*n) = A302181(n). (End)

a(13)-a(25) from Mélika Tebni, Nov 27 2024

A302184 Number of 3D walks of type abe.

Original entry on oeis.org

1, 2, 7, 26, 108, 472, 2159, 10194, 49396, 244328, 1229308, 6273896, 32410096, 169181664, 891181607, 4731912082, 25302648644, 136150941064, 736747902236, 4007011320808, 21893702201648, 120125750018656, 661630546993116, 3656966382542984, 20278320788680912, 112782556853239712

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A126120, A138547, A145847, A145867, A150500, A202814.

a := n -> 2*add(binomial(n, k)*binomial(k, k/2)*binomial(2*(n-k), n-k)/(k+2), k = 0..n, 2): seq(a(n), n = 0..25);  # Peter Luschny, Nov 30 2024

from math import comb as binomial
def a(n: int):
    return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(2*(n-k), n-k) for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Nov 30 2024

a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A000984(n-k). - Mélika Tebni, Nov 30 2024

a(12)-a(25) from Mélika Tebni, Nov 30 2024

A342770 T(n,k) is the number of rooted plane binary forests with n nodes and k trees: triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 3, 0, 1, 0, 1, 0, 5, 0, 3, 0, 1, 0, 1, 0, 0, 7, 0, 3, 0, 1, 0, 1, 0, 14, 0, 8, 0, 3, 0, 1, 0, 1, 0, 0, 22, 0, 8, 0, 3, 0, 1, 0, 1, 0, 42, 0, 24, 0, 8, 0, 3, 0, 1, 0, 1, 0, 0, 66, 0, 25, 0, 8, 0

Offset: 0

R. J. Mathar, Mar 21 2021

Multiset transform of A126120.

			See A222006 showing T(6,k).
The triangle starts (n>=0, 0<=k<=n):
  1
  0   1
  0   0   1
  0   1   0   1
  0   0   1   0   1
  0   2   0   1   0   1
  0   0   3   0   1   0   1
  0   5   0   3   0   1   0   1
  0   0   7   0   3   0   1   0   1
  0  14   0   8   0   3   0   1   0   1
  0   0  22   0   8   0   3   0   1   0   1
  0  42   0  24   0   8   0   3   0   1   0   1
  0   0  66   0  25   0   8   0   3   0   1   0   1
  0 132   0  74   0  25   0   8   0   3   0   1   0   1
  0   0 217   0  76   0  25   0   8   0   3   0   1   0   1

Cf. A222006 (row sums), A126120 (column k=1), A007595 (k=2), A046342 (k=3), A088327 (limit n->oo, row reverse).

A349935 Array read by ascending antidiagonals: A(n, k) is the n-th spin s-Catalan number, with s = k/2.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 0, 1, 5, 6, 4, 1, 1, 0, 15, 0, 5, 0, 1, 14, 36, 34, 16, 6, 1, 1, 0, 91, 0, 65, 0, 7, 0, 1, 42, 232, 364, 260, 111, 31, 8, 1, 1, 0, 603, 0, 1085, 0, 175, 0, 9, 0, 1, 132, 1585, 4269, 4600, 2666, 981, 260, 51, 10, 1, 1, 0, 4213, 0, 19845, 0, 5719, 0, 369, 0, 11, 0, 1

Offset: 1

Stefano Spezia, Dec 06 2021

			The array begins:
n\k | 1    2    3    4    5    6
----+---------------------------
  1 | 1    1    1    1    1    1 ...
  2 | 0    1    0    1    0    1 ...
  3 | 2    3    4    5    6    7 ...
  4 | 0    6    0   16    0   31 ...
  5 | 5   15   34   65  111  175 ...
  6 | 0   36    0  260    0  981 ...
  ...

William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 3.

Cf. A000012 (1st row), A059841 (2nd row).
Cf. A000027, A005043, A005891, A006003, A126120.
Cf. A349934.

T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; A[n_,k_]:=T[n,k(n+1)/2,k]-T[n,k(n+1)/2+1,k]; Flatten[Table[A[n-k+1,k],{n,12},{k,n}]]

A(n, k) = T(n, k*(n+1)/2, k) - T(n, k*(n+1)/2+1, k), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.
A(n, 1) = A126120(n+1).
A(n, 2) = A005043(n+1).
A(3, n) = A000027(n+1).
A(4, 2*n) = A005891(n).
A(5, n) = A006003(n+1).

A171509 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126931.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 33, 29, 9, 1, 110, 126, 57, 12, 1, 366, 518, 306, 94, 15, 1, 1220, 2052, 1494, 600, 140, 18, 1, 4065, 7925, 6849, 3389, 1035, 195, 21, 1, 13550, 30030, 30025, 17628, 6635, 1638, 259, 24, 1

Offset: 0

Philippe Deléham, Dec 10 2009

Equal to A053121*B^3, B = A007318.

			Triangle begins:
  1 ;
  3,1 ;
  10,6,1 ;
  33,29,9,1 ;
  110,126,57,12,1 ; ...

Cf. A053121, A054336, A096164, A171505.

Sum_{k=0..n} T(n,k)*x^k = A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -4, -3, -2, -1, 0 respectively.

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-3)^i. - Philippe Deléham, Feb 23 2012

A171839 Equal to A171368*A007318.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 3, 2, 1, 0, 0, 6, 8, 3, 1, 0, 0, 15, 22, 15, 4, 1, 0, 0, 36, 68, 52, 24, 5, 1, 0, 0, 91, 198, 191, 100, 35, 6, 1, 0, 0, 232, 586, 651, 425, 170, 48, 7, 1, 0, 0, 603, 1718, 2203, 1656, 820, 266, 63, 8, 1, 0, 0, 1585, 5047, 7285, 6299, 3591, 1435, 392, 80

Offset: 0

Philippe Deléham, Dec 19 2009

Another version of A114586.

			Triangle begins : 1 ; 0,0 ; 1,0,0 ; 1,1,0,0 ; 3,2,1,0,0 ; 6,8,3,1,0,0 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^k = A099323(n+1), A126120(n), A005043(n), A000957(n+1), A117641(n) for x = -2, -1, 0, 1, 2 respectively.

A202847 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126930.

Original entry on oeis.org

1, -1, 1, 2, -2, 1, -3, 5, -3, 1, 6, -10, 9, -4, 1, -10, 22, -22, 14, -5, 1, 20, -44, 54, -40, 20, -6, 1, -35, 93, -123, 109, -65, 27, -7, 1, 70, -186, 281, -276, 195, -98, 35, -8, 1, -126, 386, -618, 682, -541, 321, -140, 44, -9, 1

Offset: 0

Philippe Deléham, Mar 02 2013

			Triangle begins
1
-1, 1
2, -2, 1
-3, 5, -3, 1
6, -10, 9, -4, 1
-10, 22, -22, 14, -5, 1
20, -44, 54, -40, 20, -6, 1
-35, 93, -123, 109, -65, 27, -7, 1
...
Production matrix begins
x, 1
1, x, 1
1, 1, x, 1
1, 1, 1, x, 1
1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, 1, 1, x, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, x, 1
..., with x = -1.

Cf. (sequences with similar production matrix) A097609 (x=0), A033184 (x=1), A104259 (x=2), A171568 (x=3), A171589 (x=4)

T(n,k) = (-1)^(n-k)*A054336(n,k).

Sum_{k, 0<=k<=n}T(n,k)*x^k = (-1)^n*A126931(n), (-1)^n*A054341(n), A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively.

A302179 The number of 3D walks of length n in an octant returning to axis of origin.

Original entry on oeis.org

1, 1, 4, 9, 40, 120, 570, 1995, 9898, 38178, 195216, 805266, 4209084, 18239364, 96941130, 436235085, 2349133930, 10891439130, 59272544760, 281544587610, 1545550116240, 7489973640240, 41416083787260, 204122127237210, 1135679731004700, 5678398655023500, 31760915181412800, 160789633105902300

Offset: 0

N. J. A. Sloane, Apr 09 2018

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5. The sequence is type aac in Table 3.
Mélika Tebni, Fonctions de Bessel et cheminements en 3D, Dec 2024.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

C(n) = binomial(2*n, n)/(n+1); \\ A000108
f(n) = binomial(n, floor(n/2)); \\ A001405
a(n) = sum(i=0, n, if (!(i%2), sum(j=0, n-i, if (!(j%2), C(i/2)*C(j/2)*f(n-i-j)*n!/(i! * j! * (n-i-j)!))))); \\ Michel Marcus, Aug 07 2020

a(n) = Sum_{i=0..n, j=0..n-i, i,j even} A126120(i) * A126120(j) * A001405(n-i-j) * n!/(i! * j! * (n-i-j)!). - Nachum Dershowitz, Aug 06 2020

E.g.f.: (BesselI(1, 2*x)/x)^2*(BesselI(0, 2*x) + BesselI(1, 2*x)). - Mélika Tebni, Jan 06 2025

a(13)-a(27) from Nachum Dershowitz, Aug 04 2020

cp's OEIS Frontend

A138356 Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(4).

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A302182 Number of 3D walks of type abc.

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A302184 Number of 3D walks of type abe.

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A342770 T(n,k) is the number of rooted plane binary forests with n nodes and k trees: triangle read by rows.

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A349935 Array read by ascending antidiagonals: A(n, k) is the n-th spin s-Catalan number, with s = k/2.

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Mathematica

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A171509 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126931.

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A171839 Equal to A171368*A007318.

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A202847 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126930.

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A302179 The number of 3D walks of length n in an octant returning to axis of origin.

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