A098613 -id:A098613 - cp's OEIS frontend

A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.

1, 2, 5, 10, 18, 32, 55, 90, 144, 226, 346, 522, 777, 1138, 1648, 2362, 3348, 4704, 6554, 9056, 12425, 16932, 22922, 30848, 41282, 54946, 72768, 95914, 125842, 164402, 213901, 277204, 357904, 460448, 590330, 754368, 960948, 1220370, 1545306

Offset: 0

N. J. A. Sloane

The Cayley reference is actually to A079006. - Michael Somos, Feb 24 2011
In the math overflow link is a conjecture that a(n) == a(9*n + 2) (mod 4).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of 4-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019
The g.f. in the form A(x) = Sum_{k >= 0} x^(k*(k+1)) / (1 + 2*Sum_{k >= 1} (-1)^k * x^(k^2)) == Sum_{k >= 0} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + 90*x^7 + 144*x^8 + ...
G.f. = q + 2*q^5 + 5*q^9 + 10*q^13 + 18*q^17 + 32*q^21 + 55*q^25 + 90*q^29 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
joro on Math Overflow, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?.
T. Kathiravan and S. N. Fathima, On L-regular bipartitions modulo L, The Ramanujan Journal 44.3 (2017): 549-558.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 07, 1976
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001935, A022568, A079006, A082304, A098613, A127391, A127392, A210656.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548, A333374.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> [2,2,2,0] [modp(n-1,4)+1]): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
f:=(k,M) -> mul(1-q^(k*j),j=1..M); LRBP := (L,M) -> (f(L,M)/f(1,M))^2; S := L -> seriestolist(series(LRBP(L,80),q,60)); S(4); # N. J. A. Sloane, Oct 20 2019

m = 38; CoefficientList[ Series[ Product[ (1 - x^(4*k))/(1 - x^k), {k, 1, m}]^2 , {x, 0, m}], x] (* Jean-François Alcover, Sep 02 2011, after g.f. *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]) / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^2, x^2])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)

{a(n) = if( n<0, 0, polcoeff( (eta(x^4 + x * O(x^n)) / eta(x + x * O(x^n)))^2, n))};

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 / if(k%4, 1 - x^k, 1), 1 + x * O(x^n))^2, n))};

G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....
Convolution square of A001935. A079006(n) = (-1)^n a(n).
Expansion of q^(-1/4) * (1/2) * (k / k')^(1/2) in powers of q.
Euler transform of period 4 sequence [ 2, 2, 2, 0, ...].
Given g.f. A(x), then B(q) = (q * A(q^4))^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + 16*u) * (1 + 16*v) * v - u^2. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (1 + 4*u*v)^2. - Michael Somos, Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - x^k))^2.
Equals A000009 convolved with A098613. - Gary W. Adamson, Mar 24 2011
a(9*n + 2) = a(n) + 4 * A210656(3*n). - Michael Somos, Apr 02 2012
Convolution inverse is A082304. - Michael Somos, May 16 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082304. - Michael Somos, May 16 2015
Expansion of f(-x^4)^2 / f(-x)^2 = psi(x^2) / phi(-x) = psi(-x)^2 / phi(-x)^2 = psi(x)^2 / phi(-x^2)^2 = psi(x^2)^2 / psi(-x)^2 = chi(x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x)^4) = 1 / (chi(-x)^2 * chi(-x^2)^2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, May 16 2015
a(n) ~ exp(Pi*sqrt(n)) / (8*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 18 2015
G.f.: A(x) = Sum_{n >= 0} x^(n*(n+1)) / Sum_{n = -oo..oo} (-1)^n*x^(n^2). - Peter Bala, Feb 19 2021

A083906 Table read by rows: T(n, k) is the number of length n binary words with exactly k inversions.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 2, 5, 3, 4, 3, 1, 6, 4, 6, 6, 6, 2, 2, 7, 5, 8, 9, 11, 9, 7, 4, 3, 1, 8, 6, 10, 12, 16, 16, 18, 12, 12, 8, 6, 2, 2, 9, 7, 12, 15, 21, 23, 29, 27, 26, 23, 21, 15, 13, 7, 4, 3, 1, 10, 8, 14, 18, 26, 30, 40, 42, 48, 44, 46, 40, 40, 30, 26, 18, 14, 8, 6, 2, 2

Offset: 0

Alford Arnold, Jun 19 2003

There are A033638(n) values in the n-th row, compliant with the order of the polynomial.
In the example for n=6 detailed below, the orders of [6, k]_q are 1, 6, 9, 10, 9, 6, 1 for k = 0..6,
the maximum order 10 defining the row length.
Note that 1 6 9 10 9 6 1 and related distributions are antidiagonals of A077028.
A083480 is a variation illustrating a relationship with numeric partitions, A000041.
The rows are formed by the nonzero entries of the columns of A049597.
The coefficient of q^j in the Gaussian polynomial [n, m]q is the number of binary words on alphabet {0,1} of length n having m 1's and j inversions. Hence T(n, k) is the number of length n binary words with exactly k inversions. - _Geoffrey Critzer, May 14 2017
If n is even the n-th row converges to n+1, n-1, n-4, ..., 19, 13, 7, 4, 3, 1 which is A029552 reversed, and if n is odd the sequence is twice A098613. - Michael Somos, Jun 25 2017

			When viewed as an array with A033638(r) entries per row, the table begins:
. 1 ............... : 1
. 2 ............... : 2
. 3 1 ............. : 3 + q = (1) + (1+q) + (1)
. 4 2 2 ........... : 4 + 2q + 2q^2 = 1 + (1+q+q^2) + (1+q+q^2) + 1
. 5 3 4 3 1 ....... : 5 + 3q + 4q^2 + 3q^3 + q^4
. 6 4 6 6 6 2 2
. 7 5 8 9 11 9 7 4 3 1
. 8 6 10 12 16 16 18 12 12 8 6 2 2
. 9 7 12 15 21 23 29 27 26 23 21 15 13 7 4 3 1
...
The second but last row is from the sum over 7 q-polynomials coefficients:
. 1 ....... : 1 = [6,0]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,1]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,2]_q
. 1 1 2 3 3 3 3 2 1 1 ....... : 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9 = [6,3]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,4]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,5]_q
. 1 ....... : 1 = [6,6]_q

George E. Andrews, 'Theory of Partitions', 1976, page 242.

Seiichi Manyama, Rows n = 0..48, flattened
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Alexander Gruber, "The Egg:" Bizarre behavior of the roots of a family of polynomials Mathematics StackExchange Oct 04 2012
MathOverflow, Partition numbers and Gaussian binomial coefficient
Eric Weisstein, q-Binomial Coefficient, Mathworld.
Wikipedia, q-binomial
Index entries for sequences related to binary linear codes

Cf. A000025, A000034, A000041, A016116, A029552, A033638, A060546, A063746, A077028, A083479, A083480, A098613, A260460.

R:=PowerSeriesRing(Rationals(), 100);
qBinom:= func< n,k,x | n eq 0 or k eq 0 select 1 else (&*[(1-x^(n-j))/(1-x^(j+1)): j in [0..k-1]]) >;
A083906:= func< n,k | Coefficient(R!((&+[qBinom(n,k,x): k in [0..n]]) ), k) >;
[A083906(n,k): k in [0..Floor(n^2/4)], n in [0..12]]; // G. C. Greubel, Feb 13 2024

QBinomial := proc(n,m,q) local i ; factor( mul((1-q^(n-i))/(1-q^(i+1)),i=0..m-1) ) ; expand(%) ; end:
A083906 := proc(n,k) add( QBinomial(n,m,q),m=0..n ) ; coeftayl(%,q=0,k) ; end:
for n from 0 to 10 do for k from 0 to A033638(n)-1 do printf("%d,",A083906(n,k)) ; od: od: # R. J. Mathar, May 28 2009
T := proc(n, k) if n < 0 or k < 0 or k > floor(n^2/4) then return 0 fi;
if n < 2 then return n + 1 fi; 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) end:
seq(print(seq(T(n, k), k = 0..floor((n/2)^2))), n = 0..8);  # Peter Luschny, Feb 16 2024

Table[CoefficientList[Total[Table[FunctionExpand[QBinomial[n, k, q]], {k, 0, n}]],q], {n, 0, 10}] // Grid (* Geoffrey Critzer, May 14 2017 *)

{T(n, k) = polcoeff(sum(m=0, n, prod(k=0, m-1, (x^n - x^k) / (x^m - x^k))), k)}; /* Michael Somos, Jun 25 2017 */

def T(n,k): # T = A083906
    if k<0 or k> (n^2//4): return 0
    elif n<2 : return n+1
    else: return 2*T(n-1, k) - T(n-2, k) + T(n-2, k-n+1)
flatten([[T(n,k) for k in range(int(n^2//4)+1)] for n in range(13)]) # G. C. Greubel, Feb 13 2024

T(n, k) is the coefficient [q^k] of the Sum_{m=0..n} [n, m]_q over q-Binomial coefficients.
Row sums: Sum_{k=0..floor(n^2/4)} T(n,k) = 2^n.
For n >= k, T(n+1,k) = T(n, k) + A000041(k). - Geoffrey Critzer, Feb 12 2021
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A060546(n). - G. C. Greubel, Feb 13 2024
From Mikhail Kurkov, Feb 14 2024: (Start)
T(n, k) = 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) for n >= 2 and 0 <= k <= floor(n^2/4).
Sum_{i=0..n} T(n-i, i) = A000041(n+1). Note that upper limit of the summation can be reduced to A083479(n) = (n+2) - ceiling(sqrt(4*n)).
Both results were proved (see MathOverflow link for details). (End)
From G. C. Greubel, Feb 17 2024: (Start)
T(n, floor(n^2/4)) = A000034(n).
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A016116(n+1).
Sum_{k=0..(n + 2) - ceiling(sqrt(4*n))} (-1)^k*T(n - k, k) = (-1)^n*A000025(n+1) = -A260460(n+1). (End)

Edited by R. J. Mathar, May 28 2009

New name using a comment from Geoffrey Critzer by Peter Luschny, Feb 17 2024

A029552 Expansion of phi(x) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 4, 7, 13, 19, 29, 43, 62, 90, 126, 174, 239, 325, 435, 580, 769, 1007, 1313, 1702, 2191, 2808, 3580, 4539, 5735, 7216, 9036, 11278, 14028, 17383, 21474, 26448, 32471, 39759, 48550, 59123, 71829, 87053, 105249, 126975, 152858, 183623

Offset: 0

N. J. A. Sloane

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 3*x + 4*x^2 + 7*x^3 + 13*x^4 + 19*x^5 + 29*x^6 + 43*x^7 + ...
G.f. = 1/q + 3*q^23 + 4*q^47 + 7*q^71 + 13*q^95 + 19*q^119 + 29*q^143 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A083906, A098613.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, Oct 29 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^2 / QPochhammer[ q, q^2], {q, 0, n}]; (* Michael Somos, Oct 29 2013 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1) / eta(x + x * O(x^n)), n))}; /* Michael Somos, Sep 17 2004 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^3 * eta(x^4 + A)^2), n));} /* Michael Somos, Sep 17 2004 */

{a(n) = if( n<0, 0, polcoeff( sum(k=0, 2*n, prod(i=1, k, (1 -x^(2*n + 1-i)) / (1 - x^i))), n^2-n))}; /* Michael Somos, Sep 17 2004 */

Expansion of q^(1/24) * eta(q^2)^5 /(eta(q)^3 * eta(q^4)^2) in powers of q. - Michael Somos, Sep 17 2004
Euler transform of period 4 sequence [3, -2, 3, 0, ...]. - Michael Somos, Sep 17 2004
G.f. A(x) is the limit of x^(n^2) P_{2n}(1/x) where P_n(q) = Sum_{k=0..n} C(n,k;q) and C(n,k;q) is q-binomial coefficients. See A083906 for P_n. - Michael Somos, Sep 17 2004
G.f.: (1 + 2 * Sum_{k>0} x^(k^2)) / (Product_{k>0} (1 - x^k)).
a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(7/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 01 2017
Expansion of chi(x)^3/chi(-x^2) = chi(x)^2/chi(-x) = chi(-x^2)^2/chi(-x)^3 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Apr 24 2023

A143161 Expansion of chi(-x)^2 * chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 0, 3, -2, 0, 0, 4, -6, 0, 0, 7, -8, 0, 0, 13, -14, 0, 0, 19, -20, 0, 0, 29, -34, 0, 0, 43, -46, 0, 0, 62, -70, 0, 0, 90, -96, 0, 0, 126, -138, 0, 0, 174, -186, 0, 0, 239, -262, 0, 0, 325, -346, 0, 0, 435, -472, 0, 0, 580, -620, 0, 0, 769, -826, 0, 0

Offset: 0

Michael Somos, Jul 27 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - 2*x + 3*x^4 - 2*x^5 + 4*x^8 - 6*x^9 + 7*x^12 - 8*x^13 + 13*x^16 + ...
G.f. = 1/q - 2*q^5 + 3*q^23 - 2*q^29 + 4*q^47 - 6*q^53 + 7*q^71 - 8*q^77 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A029552, A098613.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ x^2, x^4], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / (eta(x^2 + A) * eta(x^4 + A)), n))};

Expansion of q^(1/6) * eta(q)^2 / (eta(q^2) * eta(q^4)) in powers of q.
Euler transform of period 4 sequence [ -2, -1, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 8^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098613. - Michael Somos, Sep 07 2015
a(4*n + 2) = a(4*n + 3) = 0.
G.f.: (Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k)))^-1.
a(4*n) = A029552(n). a(4*n + 1) = -2 * A098613(n).

A224916 Expansion of chi(x)^2 / chi(-x^2)^6 in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 7, 14, 31, 58, 112, 196, 347, 580, 966, 1554, 2485, 3872, 5993, 9102, 13719, 20384, 30068, 43836, 63481, 91048, 129763, 183448, 257839, 359862, 499583, 689312, 946416, 1292388, 1756838, 2376598, 3201557, 4293942, 5736736, 7633702, 10121408, 13370634

Offset: 0

Michael Somos, Apr 19 2013

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + 2*x + 7*x^2 + 14*x^3 + 31*x^4 + 58*x^5 + 112*x^6 + 196*x^7 + 347*x^8 + ...
q^5 + 2*q^17 + 7*q^29 + 14*q^41 + 31*q^53 + 58*q^65 + 112*q^77 + 196*q^89 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A098613, A112160.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^2 / (4 q^(1/2) QPochhammer[q]^2), {q, 0, n}]
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ q^2, q^4]^4 / QPochhammer[ q, q^2]^2, {q, 0, n}]
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2]^4 - QPochhammer[ q, q^2]^4)/ 8, {q, 0, 2 n + 1}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)))^2, n))}

Expansion of q^(-5/12) * (eta(q^4)^2 / (eta(q) * eta(q^2)))^2 in powers of q.
Expansion of psi(x^2)^2 / f(-x)^2 = 1 / (chi(-x)^2 * chi(-x^2)^4) = 1 / (chi(x)^4 * chi(-x)^6 ) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
Expansion of (chi(x)^4 - chi(-x)^4) / (8*x) in powers of x^2 where chi() is a Ramanujan theta function.
Euler transform of period 4 sequence [ 2, 4, 2, 0, ...].
G.f.: Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k))^4.
G.f.: (Sum_{k>0} x^(k^2 - k)) / (Product_{k>0} (1 - x^k))^2. - Michael Somos, Jul 04 2013
a(n) = A112160(2*n + 1) / 4.
Convolution square of A098613. - Michael Somos, Jul 04 2013
a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

A346848 Number of conjugacy classes of the symplectic group Sp(2n, 2) over the field with 2 elements.

Original entry on oeis.org

1, 3, 11, 30, 81, 198, 477, 1089, 2451, 5358, 11567, 24537, 51577, 107205, 221378, 453900, 926395, 1882152, 3812232, 7699191, 15518112, 31220991, 62733296, 125911851, 252516626, 506082933, 1013780968, 2029989807, 4063678159, 8132877129, 16274093175

Offset: 0

Jan Kristian Haugland, Aug 06 2021

nonn

Sp(2n, 2) is isomorphic to the orthogonal group O(2n+1, 2) over the field with 2 elements, and is a simple and complete group for n>=3.

			a(2)=11, and Sp(4, 2) is isomorphic to the symmetric group S_6 which has 11 conjugacy classes.

Jan Kristian Haugland, Table of n, a(n) for n = 0..1000
G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Aust. Math. Soc., 3 (1963), 1-62.

Discrete convolution of A070933 and A098613. A003923 gives the order of the group.

A385672 Irregular triangle read by rows: T(n, k) is the number of n-step walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).

Original entry on oeis.org

1, 4, 12, 2, 40, 8, 4, 124, 42, 16, 6, 2, 416, 160, 92, 28, 16, 4, 4, 1348, 678, 362, 174, 88, 34, 22, 8, 6, 2, 4624, 2548, 1624, 756, 460, 200, 156, 56, 40, 20, 12, 4, 4, 15632, 10062, 6336, 3586, 2110, 1106, 742, 388, 278, 152, 82, 46, 34, 14, 8, 6, 2

Offset: 0

Andrei Zabolotskii, Aug 04 2025

Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give 4^n.

The algebraic area is Integral y dx over the walk, which equals (Sum_{steps right} y) - (Sum_{steps left} y).

			The triangle begins:
     1
     4
    12,   2
    40,   8,   4
   124,  42,  16,   6,  2
   416, 160,  92,  28, 16,  4,  4
  1348, 678, 362, 174, 88, 34, 22, 8, 6, 2
   ...
T(3, 1) = 8: RUR (right, up, right), LUR, RDL, LDL, URU, URD, DLU, DLD.

Row lengths are A033638 = A002620 + 1.
A352838 is an analog that gives the number of closed walks.
Cf. A029552, A098613.

d = [{((0, 0), 0): 1}]
for _ in range(10):
    nd = {}
    for key, nw in d[-1].items():
        pos, ar = key
        x, y = pos
        for key in [
            ((x+1, y), ar + y),
            ((x-1, y), ar - y),
            ((x, y+1), ar),
            ((x, y-1), ar)
            ]:
            if key in nd:
                nd[key] += nw
            else:
                nd[key] = nw
    d.append(nd)
t = []
for nd in d:
    a = [0] * (max(ar for _, ar in nd) + 1)
    for key, nw in nd.items():
        _, ar = key
        if ar >= 0:
            a[ar] += nw
    t.append(a)
print(t)

It appears that T(2*n, n^2 - k) = 2 * A029552(k) for k < n and T(2*n+1, n^2+n - k) = 4 * A098613(k) for k < n.

cp's OEIS Frontend

A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.

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A083906 Table read by rows: T(n, k) is the number of length n binary words with exactly k inversions.

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A029552 Expansion of phi(x) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

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A143161 Expansion of chi(-x)^2 * chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

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A224916 Expansion of chi(x)^2 / chi(-x^2)^6 in powers of x where chi() is a Ramanujan theta function.

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A346848 Number of conjugacy classes of the symplectic group Sp(2n, 2) over the field with 2 elements.

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A385672 Irregular triangle read by rows: T(n, k) is the number of n-step walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).