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A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.

%I A067855 #79 Feb 16 2025 08:32:45
%S A067855 1,2,8,26,94,326,1196,4358,16248,60854,230184,874878,3343614,12825418,
%T A067855 49368388,190554410,737328366,2858974502,11106267880,43215101102,
%U A067855 168398785002,657070401106,2566847255572,10038191414610,39295007540748
%N A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.
%C A067855 Original name: "Squared length of sum of s_lambda^2, where s_lambda is a Schur function and lambda ranges over all partitions of n."
%C A067855 This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = 4. - _Seiichi Manyama_, Apr 22 2018
%C A067855 The symbol "|-" means "is a partition of", cf. MathWorld link and the Geloun & Ramgoolam paper. The Littlewood-Richardson coefficients allow a product of two Schur functions to be expressed as a linear combination of Schur functions of the corresponding degree. (The Schur functions symmetric in all n variables correspond to Schur polynomials of partitions extended with 0's to length n.) - _M. F. Hasler_, Jan 19 2020
%C A067855 See A070933 for similar sums of squares of Littlewood-Richardson coefficients. - _M. F. Hasler_, Jan 20 2020
%H A067855 Seiichi Manyama, <a href="/A067855/b067855.txt">Table of n, a(n) for n = 0..1000</a>
%H A067855 J. B. Geloun and S. Ramgoolam, <a href="http://arxiv.org/abs/1307.6490">Counting Tensor Model Observables and Branched Covers of the 2-Sphere</a>, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
%H A067855 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Partition.html">Partition</a>.
%H A067855 Wikipedia, <a href="https://en.wikipedia.org/wiki/Littlewood-Richardson_rule">Littlewood-Richardson rule</a>, as of Dec 18 2018.
%H A067855 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schur_polynomial">Schur polynomial</a>, as of Jan 13 2020.
%F A067855 G.f.: 1/sqrt(Product_{i >= 1} (1 - 4*x^i)).
%F A067855 Euler transform of A001868(n)/2. a(n) = Sum_{pi} Product_{m=1..n} binomial(2*p(m), p(m)), where pi runs through all nonnegative solutions of p(1) + 2*p(2) + ... + n*p(n)=n. - _Vladeta Jovovic_, Mar 25 2006
%F A067855 a(n) ~ 2^(2*n) / sqrt(c*Pi*n), where c = QPochhammer[1/4] = 0.688537537120339... - _Vaclav Kotesovec_, Apr 22 2018
%F A067855 By definition, a(n) = Sum_{mu |- 2n} c_mu^2 where Sum_{lambda |- n} s_lambda^2 = Sum_{mu |- 2n} c_mu s_mu, where s_lambda are the Schur polynomials (symmetric in 2n variables) and the sums run over all partitions of n resp. 2n. - _M. F. Hasler_, Jan 19 2020
%e A067855 For n=3 the s_lambda^2 summed over all partitions of n and decomposed into a sum of Schur functions yields
%e A067855     s(6) + 2 s(3,3) + 2 s(4,2) + s(5,1) + 2 s(2,2,2) + 2 s(3,2,1) + s(4,1,1)
%e A067855     + 2 s(2,2,1,1) + s(3,1,1,1) + s(2,1,1,1,1) + s(1,1,1,1,1,1),
%e A067855   and the sum of the squares of the coefficients {1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1} gives a(3) = 26.
%p A067855 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
%p A067855       binomial(n+n, n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
%p A067855     end:
%p A067855 a:= n-> b(n$2):
%p A067855 seq(a(n), n=0..33);  # _Alois P. Heinz_, Aug 24 2019
%t A067855 Table[Tr[(Apply[List,
%t A067855   Sum[Tr[s @@@ LRRule[\[Lambda], \[Lambda]]],
%t A067855    {\[Lambda], Partitions[n]}]] /. s[__] -> 1)^2], {n, 1, 10}];
%t A067855 (* with 'LRRule' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb - _Wouter Meeussen_, Jan 19 2020 *)
%t A067855 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 1, Binomial[n+n, n],
%t A067855      Sum[b[j, 1]*b[n - i*j, i-1], {j, 0, n/i}]]];
%t A067855 a[n_] := b[n, n];
%t A067855 Table[a[n], {n, 0, 33}] (* _Jean-François Alcover_, Jan 02 2022, after _Alois P. Heinz_ *)
%o A067855 (PARI) A067855_upto(N)=Vec(1/sqrt(prod(i=1,N-1,1-4*'x^i+O('x^N)))) \\ _M. F. Hasler_, Jan 23 2020
%Y A067855 Cf. A001868.
%Y A067855 List of partitions: A036037, A080577, A181317, A330370.
%Y A067855 Cf. A070933 (Sum_{lambda,mu,nu} (c^{lambda}_{mu,nu})^2, |mu| = |nu| = n).
%Y A067855 Cf. A003040 (maximum number of standard tableaux of the Ferrers diagrams of the partitions of n).
%K A067855 easy,nonn
%O A067855 0,2
%A A067855 _Richard Stanley_, Feb 15 2002
%E A067855 More terms from _Vladeta Jovovic_, Mar 25 2006
%E A067855 Name edited by _M. F. Hasler_ following observations by _Wouter Meeussen_, Jan 17 2020