A283674 -id:A283674

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A283716

Row n=3 of A283674.

+20
2

3, 32, 746, 19748, 531698, 14349932, 387424586, 10460369588, 282429602018, 7625597747132, 205891133143226, 5559060570749828, 150094635313776338, 4052555153086085132, 109418989131780794666, 2954312706551907440468, 79766443076876804830658

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..16.

Index entries for linear recurrences with constant coefficients, signature (32,-139,108).

FORMULA

G.f.: (3 - 64*x + 139*x^2)/((1 - x)*(1 - 4*x)*(1 - 27*x)).

a(n) = 32*a(n-1) - 139*a(n-2) + 108*a(n-3) for n > 2.

a(n) = 1 + 4^n + 27^n.

MATHEMATICA

Table[1 + 4^n + 27^n, {n, 0, 20}] (* Bruno Berselli, Mar 15 2017 *)

CoefficientList[Series[(3 - 64*x + 139*x^2)/((1 - x)*(1 - 4*x)*(1 - 27*x)), {x, 0, 17}], x] (* Indranil Ghosh, Mar 15 2017 *)

PROG

(PARI) Vec((3 - 64*x + 139*x^2)/((1 - x)*(1 - 4*x)*(1 - 27*x)) + O(x^17)) \\ Indranil Ghosh, Mar 15 2017

(PARI) a(n) = 1 + 4^n + 27^n \\ Indranil Ghosh, Mar 15 2017

(Python) def A283716(n): return 1 + 4**n + 27**n # Indranil Ghosh, Mar 15 2017

CROSSREFS

Cf. A283674.

KEYWORD

nonn,easy

AUTHOR

Seiichi Manyama, Mar 15 2017

EXTENSIONS

Extended by Bruno Berselli, Mar 15 2017

STATUS

approved

A283719

Main diagonal of A283674.

+20
2

1, 1, 17, 19748, 4295531890, 298024338096736401, 10314425731041362057808006400, 256923578002337121862310634348534055243302, 6277101735598269851830651637045695165917698976435202316054

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..26

FORMULA

a(n) ~ n^(n^2) * (1 + 1/(exp(n-1/2)*n^n)). - Vaclav Kotesovec, Mar 17 2017

MATHEMATICA

nmax = 20; Table[SeriesCoefficient[Product[1/(1 - x^k)^(k^(n*k)), {k, 1, nmax}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Mar 17 2017 *)

PROG

(Ruby)

def s(k, i)

s = 0

(1..i).each{|j| s += j ** (k * j + 1) if i % j == 0}

end

def A(k, n)

ary = [1]

s_ary = [0] + (1..n).map{|i| s(k, i)}

(1..n).each{|i| ary << (1..i).inject(0){|s, j| s + ary[-j] * s_ary[j]} / i}

ary

end

def A283719(n)

(0..n).map{|i| A(i, i)[-1]}

end

CROSSREFS

Cf. A283674.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Mar 15 2017

STATUS

approved

A283675

Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-x^j)^(j^(k*j)) in powers of x.

+10
5

1, 1, -1, 1, -1, -1, 1, -1, -4, 0, 1, -1, -16, -23, 0, 1, -1, -64, -713, -223, 1, 1, -1, -256, -19619, -64687, -2767, 0, 1, -1, -1024, -531185, -16755517, -9688545, -42268, 1, 1, -1, -4096, -14347883, -4294403215, -30499543213, -2165715003, -759008, 0, 1, -1, -16384

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Seiichi Manyama, Antidiagonals n = 0..52, flattened

FORMULA

G.f. of column k: Product_{j>=1} (1-x^j)^(j^(k*j)).

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k*d+1)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 04 2017

EXAMPLE

Square array begins:

1, 1, 1, 1, ...

-1, -1, -1, -1, ...

-1, -4, -16, -64, ...

0, -23, -713, -19619, ...

0, -223, -64687, -16755517, ...

CROSSREFS

Columns k=0..4 give A010815, A283499, A283534, A283536, A283803.

Rows n=0..1 give A000012, (-1)*A000012.

Main diagonal gives A283720.

Cf. A283674.

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Mar 14 2017

STATUS

approved

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