A067855 -id:A067855 - cp's OEIS frontend

A303360 Expansion of Product_{n>=1} ((1 + 4x^n)/(1 - 4x^n))^(1/4).

1, 2, 4, 18, 34, 166, 384, 1902, 4756, 24022, 64284, 321542, 899658, 4455690, 12888944, 63185250, 187513426, 910880550, 2759413788, 13295839638, 40967821494, 195979968882, 612569599440, 2911592648458, 9213101043936, 43538337410474, 139246245625364

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Expansion of Product_{n>=1} ((1 + 2^b*x^n)/(1 - 2^b*x^n))^(1/(2^b)): A015128 (b=0), A303346 (b=1), this sequence (b=2).

Cf. A303361, A303391, A067855, A303350, A303392.

seq(coeff(series(mul(((1+4*x^k)/(1-4*x^k))^(1/4), k = 1..n), x, n+1), x, n), n = 0..35); # Muniru A Asiru, Apr 22 2018

nmax = 30; CoefficientList[Series[Product[((1 + 4*x^k)/(1 - 4*x^k))^(1/4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
nmax = 30; CoefficientList[Series[(-3*QPochhammer[-4, x] / (5*QPochhammer[4, x]))^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 23 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+4*x^k)/(1-4*x^k))^(1/4)))

a(n) ~ c * 4^n / n^(3/4), where c = (QPochhammer[-1, 1/4] / QPochhammer[1/4])^(1/4) / Gamma(1/4) = 0.3885547372628... - Vaclav Kotesovec, Apr 23 2018

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

Original entry on oeis.org

1, 8, 40, 200, 872, 3720, 15400, 62920, 254440, 1024648, 4112680, 16483400, 66000360, 264150920, 1056903080, 4228272200, 16914393832, 67660396040, 270647139240, 1082600410440, 4330424811880, 17321748357640, 69287088965800, 277148557003720, 1108594618342760

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..1600

Cf. A261568, A246936, A067855, A303350.

Cf. A015128, A261584, A303390.

N:= 50: # for a(0)..a(N)
G:= mul((1+4*x^k)/(1-4*x^k),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Feb 13 2019

nmax = 25; CoefficientList[Series[Product[(1+4*x^k)/(1-4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n, where c = QPochhammer[-1, 1/4] / QPochhammer[1/4] = 3.9385207073365388638943873939345313401323799...

A303347 Expansion of Product_{n>=1} (1 - 4*x^n)^(1/2).

Original entry on oeis.org

1, -2, -4, -2, -6, -6, -56, -158, -612, -2070, -7228, -25238, -89646, -319466, -1150168, -4164978, -15177718, -55592614, -204617788, -756314982, -2806456898, -10450497682, -39040372248, -146273912858, -549533738952, -2069680656234, -7812908945556

Offset: 0

Seiichi Manyama, Apr 22 2018

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = 4.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Expansion of Product_{n>=1} (1 - b^2*x^n)^(1/b): A010815 (b=1), this sequence (b=2), A303348 (b=3).

Cf. A067855, A303350.

seq(coeff(series(mul((1-4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40); # Muniru A Asiru, Apr 22 2018

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-4*x^k)^(1/2)))

a(n) ~ -c * 2^(2*n-1) / (sqrt(Pi) * n^(3/2)), where c = QPochhammer[1/4]^(1/2) = 0.8297816201389011939293261374110190... - Vaclav Kotesovec, Apr 25 2018

A303349 Expansion of Product_{n>=1} 1/(1 - 9*x^n)^(1/3).

Original entry on oeis.org

1, 3, 21, 138, 1029, 7878, 62751, 508521, 4185885, 34819986, 292135143, 2467528563, 20958538377, 178846047741, 1532203949982, 13171424183184, 113562780734352, 981679181808261, 8505577753517235, 73846557073784937, 642328501788394527

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/3, g(n) = 9.

In general, if h > 1 and g.f. = Product_{k>=1} 1/(1 - h^2*x^k)^(1/h), then a(n) ~ h^(2*n) / (Gamma(1/h) * QPochhammer[1/h^2]^(1/h) * n^(1 - 1/h)). - Vaclav Kotesovec, Apr 22 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Expansion of Product_{n>=1} 1/(1 - b^2*x^n)^(1/b): A000041 (b=1), A067855 (b=2), this sequence (b=3).

Cf. A303348.

seq(coeff(series(mul(1/(1-9*x^k)^(1/3), k = 1..n), x, n+1), x, n), n=0..25); # Muniru A Asiru, Apr 22 2018

nmax = 20; CoefficientList[Series[Product[1/(1 - 9*x^k)^(1/3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

a(n) ~ c * 3^(2*n) / n^(2/3), where c = 1 / (Gamma(1/3) * QPochhammer[1/9]^(1/3)) = 0.390040743840141117482137514... - Vaclav Kotesovec, Apr 22 2018

A303352 Expansion of Product_{n>=1} 1/(1 + 4*x^n)^(1/2).

Original entry on oeis.org

1, -2, 4, -18, 66, -230, 832, -3118, 11764, -44374, 168476, -643974, 2470506, -9503946, 36666736, -141824034, 549717490, -2134650662, 8303024092, -32343942934, 126161860886, -492703658930, 1926278860624, -7538530620746, 29529208903872, -115766389203370

Offset: 0

Seiichi Manyama, Apr 22 2018

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = -4.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Expansion of Product_{n>=1} 1/(1 + b^2*x^n)^(1/b): A081362 (b=1), this sequence (b=2), A303353 (b=3).

Cf. A067855, A303350.

seq(coeff(series(mul(1/(1+4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40); # Muniru A Asiru, Apr 22 2018

nmax = 30; CoefficientList[Series[Product[1/(1 + 4*x^k)^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)

a(n) ~ c * (-4)^n / sqrt(Pi*n), where c = 1 / QPochhammer[-1/4]^(1/2) = 0.91806413264267465793225216525758518... - Vaclav Kotesovec, Apr 25 2018

A330985 Irregular table read by rows in which row n gives the Littlewood-Richardson coefficients for the square of the symmetric Schur function corresponding to the n-th partition listed in A036036 (colexicographic order).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1

Offset: 1

M. F. Hasler, Jan 21 2020

Not only the rows but also the coefficients in each row are listed in one-to-one correspondence with the partitions as listed in the corresponding row of A036036.
The length of row n in this table equals A000041(2|lambda|), the number of partitions of 2|lambda|, where |lambda| is the sum of parts of the n-th partition listed in A036036. There are A000041(k) rows of length A000041(2k), k >= 1.
The graded colexicographic order is also known as "Abramovitz-Stegun" or better Hindenburg order, cf. Luschny link. (This is the lexicographic order of the partitions padded with '0's to length |lambda| and with parts in increasing order, see column "Ref Colex" on the OEIS Wiki page.)
To each partition lambda is associated a Schur polynomial s_lambda through Jacobi's bialternant formula. The Littlewood-Richardson coefficients are the structure constants in the ring of symmetric functions w.r.t. the basis of Schur functions, i.e., they are the coefficients of products s_mu*s_nu written as linear combinations of the Schur functions s_lambda of degree |lambda| = |mu| + |nu|. (To get this well-defined in terms of symmetric functions, we must consider the polynomials s_mu, s_nu also in |lambda| variables.) This table considers the diagonal of this multiplication table, corresponding to squares of Schur polynomial functions.
Sequence A067855 gives the sum of squares of the coefficients of Sum_{mu |- n} s_mu^2. This corresponds to taking the sum, as vectors, of rows of equal length (equivalent to equal |mu|), and then taking the Euclidean norm squared. For example, for mu |- 2 <=> |mu| = 2, take the sum of rows 2 and 3, to get (1, 1, 2, 1, 1), with sum of squares equal to 8 = A067855(2).
It is known that L-R coefficients for products of "rectangular" partitions contain only 0's and 1's (Okada 1998), therefore rows 5, 8, 10, ... are the first rows that may have terms > 1.

			The 4th partition listed in A036036 is (1,2); the Schur function (s[1,2])^2 is equal to 0*s[6] + 0*s[1,5] + 1*s[2,4] + 1*s[3,3] + 1*s[1,1,4] + 2*s[1,2,3] + 1*s[2,2,2] + 1*s[1,1,1,3] + 1*s[1,1,2,2] + 0*s[1,1,1,1,2] + 0*s[1,1,1,1,1,1], therefore the 4th row is (0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0).
The table starts:
   n | partition mu | 2|mu| | coefficients of (s_mu)^2
  ---+--------------+-------+---------- ----------------
   1 |     (1)      |   2   | (1, 1)
   2 |     (2)      |   4   | (1, 1, 1, 0, 0)
   3 |    (1,1)     |   4   | (0, 0, 1, 1, 1)
   4 |     (3)      |   6   | (1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0)
   5 |    (1,2)     |   6   | (0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0)
   6 |   (1,1,1)    |   6   | (0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1)

Peter Luschny, Counting with partitions.
OEIS Wiki, Orderings of partitions (a comparison).
Soichi Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, Journal of Algebra, vol. 205, no 2, 1998, pp. 337-367. DOI: 10.1006/jabr.1997.7408.
Wikipedia, Littlewood-Richardson rule, as of Dec 18 2018.
Wikipedia, Schur polynomial, as of Jan 13 2020.

Cf. A000041 (partition numbers), A036036 (partitions in colex order).

Cf. A067855 (square of the L2-norm of the vector sum of rows of equal length).

s(p,x=eval([Str("'x"i)|i<-[1..#p]]))={my(J(p)=matdet(matrix(#p,#p, i,j, x[i]^p[j]))); J(Vec(p)+[0..#p-1])/J([0..#p-1])} \\ Schur polynomial corresponding to partition p with p(1) <= ... <= p(n) (otherwise the result differs!).
lead(P,m=1)={while(poldegree(P),m*=variable(P)^poldegree(P);P=pollead(P));m} \\ leading monomial of the polynomial P
lcoef(P)={while(poldegree(P),P=pollead(P));P} \\ coeff. of leading monomial
Schur_index(n,B=Map())={forpart(p=n,mapput(B,lead(s(p)),p));B} \\ Initialize the index {leading monomial => partition}
/* The following computes the row corresponding to partition p, but not very efficiently: it requires lots of memory for |mu| >= 4 (<=> |lambda| >= 8). */
c(p, n=vecsum(Vec(p))*2, B=Schur_index(n))={my(S=s(vecsort(Vec(p,-n)))^2, C=Map()); while(S, my(c); mapput(C, p=mapget(B,lead(S)), c=lcoef(S)); S-=c*s(Vec(p,-n)); if(default(debug), printf("%+d s%d ",c,Vec(p)))); [iferr(mapget(C,p),E,0) | p<-partitions(n)]} \\ If debug>0 (\g1), prints the s_lambda when found in s_p^2.
A330985=concat([c([1]),c([2]),c([1,1]),c([3]),c([2,1]),c([1,1,1])])
A330985_row(n)=for(k=1,oo,(0

s_mu^2 = Sum_{k=1..A000041(2|mu|)} T(n,k)*s_{p(k,2|mu|)}, where mu is the n-th partition listed in A036036, p(k,2|mu|) is the k-th partition in row 2|mu| of A036036, and s_mu, s_p are the Schur functions (or polynomials in 2|mu| variables) associated to the partitions mu resp. p.

A303392 Expansion of Product_{k>=1} ((1 + 4x^k) / (1 - 4x^k))^(1/2).

Original entry on oeis.org

1, 4, 12, 52, 156, 612, 2028, 7892, 27324, 107396, 384844, 1520436, 5566876, 22069796, 81990252, 325707348, 1222582268, 4862950020, 18395472460, 73233825524, 278700724764, 1110232691108, 4245596648876, 16920914168148, 64963831455996, 259012955299396

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Cf. A261568, A246936, A067855, A303350, A303391, A303360.

nmax = 30; CoefficientList[Series[Product[((1+4*x^k)/(1-4*x^k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[(-3*QPochhammer[-4, x] / (5*QPochhammer[4, x]))^(1/2), {x, 0, nmax}], x]

a(n) ~ sqrt(c) * 4^n / sqrt(Pi*n), where c = QPochhammer[-1, 1/4]/QPochhammer[1/4] = 3.9385207073365388638943873939345313401323799...

A361286 Total over all partitions lambda of n, of factors of s_lambda in the skew Schur function s_( nu/lambda ) with (s_lambda)^2 = Sum( C(nu, lambda, lambda) s_nu ).

Original entry on oeis.org

1, 2, 6, 18, 50, 138, 430, 1242, 3666, 10938, 34598, 108098, 338634, 1058370

Offset: 0

Wouter Meeussen, Mar 07 2023

All the terms for n >= 1 so far are twice an odd integer.
In terms of Young diagrams, this counts how many original copies one gets by first adding n boxes and then removing n boxes while maintaining an allowed Young diagram shape.
Also a(n) is the total over all partitions n of the multiplicities squared, partition by partition, in the LR-expansion of (s_lambda |- n)^2. Notice that this is different from A067855 where the multipliciteis are first summed over all lambda |-n, and finally squared, then summed.

			For n=3,
    {3} -> 4 s_{3} + 2 s_{2,1}
    {2,1} -> 4 s_{3} + 10 s_{2,1} + 4 s_{1,1,1} and
    {1,1,1} -> 2 s_{2,1} + 4 s_{1,1,1}
so a(3) = 4 + 10 + 4 = 18.
Also,
s(3)^2 -> s(6)+s(3;3)+s(4;2)+s(5,1) -> {1,1,1,1} ->{1,1,1,1} ->4
  s(2;1)^2 ->s(4;2)+s(4;1;1)+s(3;3)+2 s(3;2;1)+s(3;1;1;1)+s(2;2;2)+s(2;2;1;1)
         -> {1,1,1,2,1,1,1} -> {1,1,1,4,1,1,1} -> 10
s(1;1;1)^2 -> s(2;2;2)+s(2;2;1;1)+s(2;1;1;1;1)+s(1^6) ->{1,1,1,1} ->{1,1,1,1} ->4

Cf. A067855, A322210.

(* with 'LRRule' and 'skewschur' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb *)
Tr/@ Table[Coefficient[
  Total[skewschur[#, \[Lambda], n] & /@
    LRRule[\[Lambda], \[Lambda]]], ss[\[Lambda], n] ], {n,
  13}, {\[Lambda], Partitions[n]}];
also Table[Total[
  Table[Map[Last, Tally[LRRule[\[Lambda], \[Lambda]]] ]^2, {\[Lambda],
     Partitions[n]}], 2], {n, 13}];

A329155 Expansion of Product_{k>=1} 1 / (1 - 2x^k - 3x^(2*k))^(1/2).

Original entry on oeis.org

1, 1, 4, 9, 27, 67, 193, 515, 1462, 4070, 11588, 32898, 94389, 271017, 782401, 2263002, 6565987, 19086043, 55597255, 162207806, 473992799, 1386875848, 4062919108, 11915397853, 34979609583, 102781548770, 302259362326, 889566748760, 2619915414564, 7721166976185

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

nonn

Cf. A002426, A067855, A081362, A242587.

nmax = 29; CoefficientList[Series[Product[1/(1 - 2 x^k - 3 x^(2 k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Exp[Sum[Sum[(3^d + (-1)^d)/d, {d, Divisors[k]}] x^k/2, {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 - x^(2*k - 1)) / (1 - 3*x^k))^(1/2).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (3^d + (-1)^d) / d ) * x^k / 2).
G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A002426 (central trinomial coefficients).
a(n) ~ c * 3^(n + 1/2) / (2*sqrt(Pi*n)), where c = sqrt(Product_{k>=2} 1/((1 - 1/3^(k-1))*(1 + 1/3^k))) = sqrt(8 / (3 * QPochhammer[-1, 1/3] * QPochhammer[1/3])) = 1.23332761652608605487734981242239445... - Vaclav Kotesovec, Nov 07 2019

A329806 Expansion of Product_{k>=1} 1 / (1 - 6x^k + x^(2k))^(1/2).

Original entry on oeis.org

1, 3, 16, 75, 385, 1971, 10473, 56139, 305394, 1674198, 9245506, 51325206, 286210243, 1601822505, 8992732043, 50619114252, 285583525237, 1614439389711, 9142794839933, 51858472602546, 294559269778199, 1675240507900632, 9538522900076376, 54367531265208579, 310179797595736539

Offset: 0

Ilya Gutkovskiy, Nov 21 2019

nonn

Cf. A001850, A067855.

nmax = 24; CoefficientList[Series[Product[1/(1 - 6 x^k + x^(2 k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[Sum[Sum[d (6 - x^d)^(k/d), {d, Divisors[k]}] x^k/(2 k), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d * (6 - x^d)^(k/d) ) * x^k / (2*k)).
G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A001850 (central Delannoy numbers).
a(n) ~ sqrt(2) * (1 + sqrt(2))^(2*n - 1/2) / (c * sqrt(Pi*n)), where c = QPochhammer[1/(1 + sqrt(2))^2] = 0.799142925985081767883272500537236047... - Vaclav Kotesovec, Nov 21 2019

cp's OEIS Frontend

A303360 Expansion of Product_{n>=1} ((1 + 4*x^n)/(1 - 4*x^n))^(1/4).

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A303391 Expansion of Product_{k>=1} (1 + 4*x^k)/(1 - 4*x^k).

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A303347 Expansion of Product_{n>=1} (1 - 4*x^n)^(1/2).

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A303349 Expansion of Product_{n>=1} 1/(1 - 9*x^n)^(1/3).

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A303352 Expansion of Product_{n>=1} 1/(1 + 4*x^n)^(1/2).

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A330985 Irregular table read by rows in which row n gives the Littlewood-Richardson coefficients for the square of the symmetric Schur function corresponding to the n-th partition listed in A036036 (colexicographic order).

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A303392 Expansion of Product_{k>=1} ((1 + 4*x^k) / (1 - 4*x^k))^(1/2).

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A361286 Total over all partitions lambda of n, of factors of s_lambda in the skew Schur function s_( nu/lambda ) with (s_lambda)^2 = Sum( C(nu, lambda, lambda) s_nu ).

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A303360 Expansion of Product_{n>=1} ((1 + 4x^n)/(1 - 4x^n))^(1/4).

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

A303392 Expansion of Product_{k>=1} ((1 + 4x^k) / (1 - 4x^k))^(1/2).

A329155 Expansion of Product_{k>=1} 1 / (1 - 2x^k - 3x^(2*k))^(1/2).

A329806 Expansion of Product_{k>=1} 1 / (1 - 6x^k + x^(2k))^(1/2).