A294653 -id:A294653 - cp's OEIS frontend

A292312 Expansion of Product_{k>=1} (1 - k^k*x^k).

1, -1, -4, -23, -229, -2761, -42615, -758499, -15702086, -365588036, -9516954786, -273061566624, -8575969258607, -292418459301779, -10762887030763337, -425243370397722674, -17953905924215881215, -806666656048846472309

Offset: 0

Seiichi Manyama, Sep 14 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Column k=1 of A294653.

Cf. A023882, A265949.

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 - k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

seq(coeff(series(mul((1-k^k*x^k),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

terms = 18; CoefficientList[Product[(1 - k^k*x^k), {k, 1, terms}] + O[x]^(terms), x] (* Jean-François Alcover, Nov 11 2017 *)

{a(n) = polcoeff(prod(k=1, n, 1-k^k*x^k+x*O(x^n)), n)}

Convolution inverse of A023882.
a(n) ~ -n^n * (1 - exp(-1)/n - (exp(-1)/2 + 4*exp(-2))/n^2). - Vaclav Kotesovec, Sep 14 2017
a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294645(k)*a(n-k) for n > 0. - Seiichi Manyama, Nov 09 2017

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -8, -1, 1, -1, -32, -73, 0, 1, -1, -128, -2155, -927, 4, 1, -1, -512, -58921, -259701, -13969, 4, 1, -1, -2048, -1593811, -67045719, -48496253, -254580, 7, 1, -1, -8192, -43044673, -17178209325, -152513227585, -13001952944, -5288596, 3

Offset: 0

Seiichi Manyama, Nov 09 2017

			Square array begins:
    1,      1,         1,             1,                1, ...
   -1,     -1,        -1,            -1,               -1, ...
   -2,     -8,       -32,          -128,             -512, ...
   -1,    -73,     -2155,        -58921,         -1593811, ...
    0,   -927,   -259701,     -67045719,     -17178209325, ...
    4, -13969, -48496253, -152513227585, -476819162106101, ...

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

Columns k=0..2 give A073592, A294809, A294953.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A004171.
Cf. A294653, A294950.

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j)) * A(n-j,k) for n > 0.

A294699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j(k*j) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -16, 0, 1, -1, -256, -713, 0, 1, -1, -4096, -531185, -64711, 1, 1, -1, -65536, -387416393, -4294405135, -9688521, 0, 1, -1, -1048576, -282429470945, -281474581032631, -95363000655153, -2165724176, 1, 1

Offset: 0

Seiichi Manyama, Nov 07 2017

			Square array begins:
    1,      1,           1,                1, ...
   -1,     -1,          -1,               -1, ...
   -1,    -16,        -256,            -4096, ...
    0,   -713,     -531185,       -387416393, ...
    0, -64711, -4294405135, -281474581032631, ...

Seiichi Manyama, Antidiagonals n = 0..38, flattened

Columns k=0..1 give A010815, A294704.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A294583, A294653, A294756.

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

A294758 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 32, 5, 1, 1, 65, 746, 304, 7, 1, 1, 257, 19748, 66538, 3537, 11, 1, 1, 1025, 531698, 16801060, 9843827, 52010, 15, 1, 1, 4097, 14349932, 4295564530, 30535638897, 2188210276, 895397, 22, 1, 1, 16385, 387424586, 1099527026284, 95371863254051, 101591953731770, 680615495493, 18016416, 30

Offset: 0

Seiichi Manyama, Nov 08 2017

			Square array begins:
   1,   1,     1,        1, ...
   1,   1,     1,        1, ...
   2,   5,    17,       65, ...
   3,  32,   746,    19748, ...
   5, 304, 66538, 16801060, ...

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

Columns k=0..1 give A000041, A023882.
Rows n=0-1 give A000012.
Cf. A294653.

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

cp's OEIS Frontend

A292312 Expansion of Product_{k>=1} (1 - k^k*x^k).

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A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.

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A294699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j(k*j) in powers of x.

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A294758 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j) in powers of x.

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cp's OEIS Frontend

A292312 Expansion of Product_{k>=1} (1 - k^k*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j)^j in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j)^j(k*j) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294758 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(k*j)*x^j) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.

A294699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j(k*j) in powers of x.

A294758 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j) in powers of x.