A181119 -id:A181119 - cp's OEIS frontend

A182188 A sequence of row differences for table A182119.

1, -1, -11, -69, -407, -2377, -13859, -80781, -470831, -2744209, -15994427, -93222357, -543339719, -3166815961, -18457556051, -107578520349, -627013566047, -3654502875937, -21300003689579

Offset: 0

Kenneth J Ramsey, Apr 17 2012

This is a list of row differences corresponding to a difference of 1 in table A182119, column 0. If A181119(k+1,0) - A182119(k,0) = 1, then a(n) = A182119(k+1,n) - A182119(k,n).

If p is a prime of the form 8*n +- 3, then a(p) == 3 (mod p). If p is a prime of the form 8*n +- 1, then a(p) == -1 (mod p).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A000129, A001109, A001333, A001542, A005409, A182189, A182190, A182119.

m = 13;n = 3; c = 0;
list3 = Reap[While[c < 22, t = 6 n - m - 4; Sow[t];m = n; n = t;c++]][[2,1]]
Table[1 -Fibonacci[2*n, 2], {n,0,40}] (* G. C. Greubel, May 24 2021 *)

[1 - lucas_number1(2*n,2,-1) for n in (0..40)] # G. C. Greubel, May 24 2021

a(n) = 6*a(n-1) - a(n-2) - 4. [corrected by Klaus Purath, Mar 19 2021]
a(n) = -(A182189(n-1) + 2*A182190(n-1)).
a(n) = 2 - A182189(n).
From Klaus Purath, Mar 19 2021: (Start)
a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3).
a(n) = (-1)*Sum_{i=1..2*n-1} A001333(i) for n > 0.
a(n) = 1 - A001542(n) for n > 0.
a(n) = 1 - 2*A001109(n) for n > 0.
a(n) = (-1)*A005409(2*n) for n > 0. (End)
G.f.: (1 - 8*x + 3*x^2)/((1-x)*(1-6*x+x^2)). - Chai Wah Wu, Apr 08 2021
a(n) = 1 - Pell(2*n), where Pell(n) = A000129(n). - G. C. Greubel, May 24 2021

A066931 Number of ways to tile hexagon of edge n with diamonds of side 1, not counting rotations and reflections as different.

Original entry on oeis.org

1, 1, 6, 113, 20174, 22306955, 123222909271, 3283834214485890, 421263391026827547540, 260028731850596651411721718, 772086476515163830856527013278243, 11025620741283840573496993339545350520150, 757129347300072898736973484532998417574513923224

Offset: 0

R. K. Guy, Feb 05 2002

nonn

R. K. Guy and D. J. Reble, Illustration of initial terms
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.

Cf. A008793.

From Peter J. Taylor, Jun 17 2015: (Start)
For odd n, a(n) = A008793(n)/12 + A049505(n)/4 + A006366(n)/6.
For even n, a(n) = A008793(n)/12 + A049505(n)/4 + A006366(n)/6 + A181119(n/2)/4 + A259049(n/2)/12 + A049503(n/2)/6.
See Taylor link.
(End)

One more term from Don Reble, Feb 07 2002

More terms from Peter J. Taylor, Jun 17 2015

A278289 Number of standard Young tableaux of skew shape (2n-1,2n-2,...,2,1)/(n-1,n-2,..,2,1).

Original entry on oeis.org

1, 1, 16, 101376, 1190156828672, 68978321274090930831360, 40824193474825703180733027309531955200, 440873872874088459550341319780612789503586208384381091840, 140992383930585613207663170866505518985873138480180692888967131590224605582721024

Offset: 0

Alejandro H. Morales, Nov 16 2016

nonn

			For n = 3 there are a(2) = 16 standard tableaux of shape (3,2,1)/(1).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Corollary 7.16.3.

Alejandro H. Morales, Table of n, a(n) for n = 0..22
A. H. Morales, I. Pak and G. Panova, Asymptotics of the number of standard Young tableaux of skew shape, arXiv:1610.07561 [math.CO], 2016; European Journal of Combinatorics, Vol 70 (2018).
A. H. Morales, I. Pak and G. Panova, Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications, arXiv:1610.04744 [math.CO], 2016; SIAM Journal of Discrete Mathematics, Vol 31 (2017).
A. H. Morales, I. Pak and M. Tassy, Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings, arXiv:1805.00992 [math.CO], 2018.
A. H. Morales and D. G. Zhu, On the Okounkov--Olshanski formula for standard tableaux of skew shapes, arXiv:2007.05006 [math.CO], 2020.
H. Naruse, Schubert calculus and hook formula, talk slides at 73rd Sém. Lothar. Combin., Strobl, Austria, 2014.
I. Pak, Skew shape asymptotics, a case-based introduction, 2020.
Jay Pantone, File with list of n, a(n) for n = 0..438 (warning: file size is 100MB)

Cf. A005118; for even n the number of terms in Naruse hook length formula is given by A181119 (Corollary 8.1 in arXiv:1610.04744).

a:=proc(k) local lam,mu;
lam:=[seq(2*k-i,i=1..2*k-1)];
mu:=[seq(k-i,i=1..k-1),seq(0,i=1..k)];
factorial(binomial(2*k,2)-binomial(k,2))*LinearAlgebra:-Determinant(Matrix(2*k-1, 2*k-1,(i,j)->`if`(lam[i]-mu[j]-i+j<0,0,1/factorial(lam[i]-mu[j]-i+j))));
end proc:
seq(a(n),n=0..5);

a(n) = ((3*n^2-n)/2)!*det(1/(lambda[i]-mu[j]-i+j)!), where lambda = (2*n-1,2*n-2,...,1) and mu = (n-1,n-2,...,1,0...,0).

There is a constant c such that log(a(k)) = n*log(n)/2 + c*n + o(n) where n = k*(3*k-1)/2 goes to infinity and -0.2368 <= c <= -0.1648. [updated by Alejandro H. Morales, Aug 29 2020]

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A182188 A sequence of row differences for table A182119.

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A066931 Number of ways to tile hexagon of edge n with diamonds of side 1, not counting rotations and reflections as different.

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A278289 Number of standard Young tableaux of skew shape (2n-1,2n-2,...,2,1)/(n-1,n-2,..,2,1).

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