A180602 -id:A180602 - cp's OEIS frontend

A086459 Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1, 2, 4, 8, ..., 2^(n-1)) right.

1, -3, 49, -3375, 923521, -992436543, 4195872914689, -70110209207109375, 4649081944211090042881, -1227102111503512992112190463, 1291749870339606615892191271170049, -5429914198235566686555216227881787109375

Offset: 1

T. D. Noe, Jul 21 2003

Note that if the rows are rotated left instead of right, the sign of the terms for which n = 0 or 3 (mod 4) is reversed. The n eigenvalues of these circulant matrices lie on the circle of radius 2(2^n - 1)/3 centered at x = (2^n - 1)/3, y = 0. This sequence can be generalized to bases other than 2 and similar results are true.

			a(3) = determinant of the matrix ((1,2,4),(4,1,2),(2,4,1)) = 49. [Corrected by _T. D. Noe_, Jan 22 2008]

Richard Bellman, Introduction to Matrix Analysis, Second Edition, SIAM, 1970, pp. 242-3.
Philip J. Davis, Circulant Matrices, Second Edition, Chelsea, 1994.

Eric Weisstein's World of Mathematics, Circulant Matrix

Cf. A048954 (circulant of binomial coefficients), A052182 (circulant of natural numbers), A066933 (circulant of prime numbers).

Cf. A180602 (unsigned, offset 0). [Paul D. Hanna, Sep 11 2010]

restart:with (combinat):a:=n->mul(-stirling2(n,2), j=3..n): seq(a(n), n=2..19); # Zerinvary Lajos, Jan 01 2009

Table[x=2^Range[0, n-1]; m=Table[RotateRight[x, i-1], {i, n}]; Det[m], {n, 12}]

a(n) = (-2^n + 1)^(n-1).

See formulas in A180602, an unsigned version of this sequence with offset 0. [Paul D. Hanna, Sep 11 2010]

A180606 G.f.: A(x) = exp( Sum_{n>=1} (2^n-1)^(n-1)*x^n/n ).

Original entry on oeis.org

1, 1, 2, 18, 862, 185582, 165592644, 599576207236, 8764375895813558, 516573425446007525398, 122710727732550268600238492, 117431929105754130321446873061820

Offset: 0

Paul D. Hanna, Sep 26 2010

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 18*x^3 + 862*x^4 + 185582*x^5 +...
log(A(x)) = x + 3*x^2/2 + 7^2*x^3/3 + 15^3*x^4/4 + 31^4*x^5/5 +...

Cf. A180602.

{a(n)=polcoeff(exp(sum(m=1,n,(2^m-1)^(m-1)*x^m/m)+x*O(x^n)),n)}

A202990 E.g.f: Sum_{n>=0} 3^n * 2^(n^2) * exp(-22^nx) * x^n/n!.

Original entry on oeis.org

1, 4, 100, 10648, 4477456, 7339040224, 47045881000000, 1186980379913527168, 118530511097526559703296, 47035767668340696232372862464, 74367598058372171073462490000000000, 469253945833810205185008441288962454059008

Offset: 0

Paul D. Hanna, Dec 26 2011

nonn

E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.

O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

			E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 10648*x^3/3! + 4477456*x^4/4! +..
By the series identity, the e.g.f.:
A(x) = exp(-2*x) + 3*2*exp(-2*2*x)*x + 3^2*2^4*exp(-2*2^2*x)*x^2/2! + 3^3*2^9*exp(-2*2^3*x)*x^3/3! +...
expands into:
A(x) = 1 + 4*x + 10^2*x^2/2! + 22^3*x^3/3! + 46^4*x^4/4! + 94^5*x^5/5! +...+ (3*2^n-2)^n*x^n/n! +...

Cf. A180602, A165327, A202989, A060613, A055601.

Table[(3*2^n-2)^n,{n,0,12}] (* Harvey P. Dale, Jul 16 2023 *)

PARI
```
{a(n, q=2, m=3, b=-2)=(m*q^n + b)^n}
```

{a(n, q=2, m=3, b=-2)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}

{a(n, q=2, m=3, b=-2)=polcoeff(sum(k=0, n, m^k*q^(k^2)*x^k/(1-b*q^k*x+x*O(x^n))^(k+1)), n)}

a(n) = (3*2^n - 2)^n.

O.g.f.: Sum_{n>=0} 3^n * 2^(n^2) * x^n/(1 + 2*2^n*x)^(n+1).

A202991 E.g.f: Sum_{n>=0} 3^(n^2) * exp(-23^nx) * x^n/n!.

Original entry on oeis.org

1, 1, 49, 15625, 38950081, 812990017201, 147640825624179889, 237771659632917369765625, 3425319186561140076700951192321, 443021141828981570872668681812345111521, 515202988063835984513918825523304657054713360049

Offset: 0

Paul D. Hanna, Dec 26 2011

nonn

E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.

O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

			E.g.f.: A(x) = 1 + x + 49*x^2/2! + 15625*x^3/3! + 38950081*x^4/4! +...
By the series identity, the g.f.:
A(x) = exp(-2*x) + 3*exp(-2*3*x)*x + 3^4*exp(-2*3^2*x)*x^2/2! + 3^9*exp(-2*3^3*x)*x^3/3! + 3^16*exp(-2*3^4*x)*x^4/4! +...
expands into:
A(x) = 1 + x + 7^2*x^2/2! + 25^3*x^3/3! + 79^4*x^4/4! + 241^5*x^5/5! +...+ (3^n-2)^n*x^n/n! +...

Cf. A180602, A165327, A202990, A202989, A060613, A055601.

PARI
```
{a(n, q=3, m=1, b=-2)=(m*q^n + b)^n}
```

{a(n, q=3, m=1, b=-2)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}

{a(n, q=3, m=1, b=-2)=polcoeff(sum(k=0, n, m^k*q^(k^2)*x^k/(1-b*q^k*x+x*O(x^n))^(k+1)), n)}

a(n) = (3^n - 2)^n.

O.g.f.: Sum_{n>=0} 3^(n^2)*x^n/(1 + 2*3^n*x)^(n+1).

cp's OEIS Frontend

A086459 Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1, 2, 4, 8, ..., 2^(n-1)) right.

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A180606 G.f.: A(x) = exp( Sum_{n>=1} (2^n-1)^(n-1)*x^n/n ).

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A202990 E.g.f: Sum_{n>=0} 3^n * 2^(n^2) * exp(-2*2^n*x) * x^n/n!.

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A202991 E.g.f: Sum_{n>=0} 3^(n^2) * exp(-2*3^n*x) * x^n/n!.

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A202990 E.g.f: Sum_{n>=0} 3^n * 2^(n^2) * exp(-22^nx) * x^n/n!.

A202991 E.g.f: Sum_{n>=0} 3^(n^2) * exp(-23^nx) * x^n/n!.