A160573 -id:A160573 - cp's OEIS frontend

A151687 G.f.: x + x^2 * Product_{n>=0} (1 + x^(2^n-1) + x^(2^n)).

0, 1, 2, 3, 3, 3, 5, 6, 4, 3, 5, 6, 6, 8, 11, 10, 5, 3, 5, 6, 6, 8, 11, 10, 7, 8, 11, 12, 14, 19, 21, 15, 6, 3, 5, 6, 6, 8, 11, 10, 7, 8, 11, 12, 14, 19, 21, 15, 8, 8, 11, 12, 14, 19, 21, 17, 15, 19, 23, 26, 33, 40, 36, 21, 7, 3, 5, 6, 6, 8, 11, 10, 7, 8, 11, 12, 14, 19, 21, 15, 8, 8

Offset: 0

N. J. A. Sloane, Jun 03 2009

nonn

Apart from initial terms and offset, same as A160573, but has a slightly nice recurrence.

Rows of triangle in A118977 converge to this.

G:= x + x^2 * mul( 1 + x^(2^n-1) + x^(2^n), n=0..20);

a(0) = 0, a(2^k) = k+1; for n >= 3, if n = 2^i + j, 1 <= j < 2^i, a(n) = a(j) + a(j+1).

A151683 Irregular triangle read by rows: row n (n>=0) gives binomial(wt(n+k),k), k >= 0, up to the point where the terms are all zeros (wt() = A000120()).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 3, 1, 1, 1, 2, 3, 1, 1, 1, 3, 3, 4, 1, 3, 6, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 3, 1, 1, 1, 2, 3, 1, 1, 1, 3, 3, 4, 1, 3, 6, 1, 4, 1, 1, 1, 2, 3, 1, 1, 1, 3, 3, 4, 1, 3, 6, 1, 1, 1, 4, 3, 4, 1, 1, 1, 3

Offset: 0

N. J. A. Sloane, Jun 01 2009

Suggested by Hagen von Eitzen's formula for A160573.

			The rows for n = 0 .. 36 are:
. 1, 1,
. 1, 1, 1,
. 1, 2,
. 1, 1, 1,
. 1, 2, 1, 1,
. 1, 2, 3,
. 1, 3,
. 1, 1, 1,
. 1, 2, 1, 1,
. 1, 2, 3,
. 1, 3, 1, 1,
. 1, 2, 3, 1, 1,
. 1, 3, 3, 4,
. 1, 3, 6,
. 1, 4,
. 1, 1, 1,
. 1, 2, 1, 1,
. 1, 2, 3,
. 1, 3, 1, 1,
. 1, 2, 3, 1, 1,
. 1, 3, 3, 4,
. 1, 3, 6,
. 1, 4, 1, 1,
. 1, 2, 3, 1, 1,
. 1, 3, 3, 4,
. 1, 3, 6, 1, 1,
. 1, 4, 3, 4, 1, 1,
. 1, 3, 6, 4, 5,
. 1, 4, 6, 10,
. 1, 4, 10,
. 1, 5,
. 1, 1, 1,
. 1, 2, 1, 1,
. 1, 2, 3,
. 1, 3, 1, 1,
. 1, 2, 3, 1, 1,
. 1, 3, 3, 4,
...

Row sums are A160573.

A151684 Let c(n) = x^(2^n-1)(1-x^(2^n)), g(n) = 1 + x^(2^n-1) + x^(2^n), h(n) = Product_{i=1..n} g(i); then use g.f. (1+2x) - Sum_{n>=1} c(n)/h(n).

Original entry on oeis.org

1, 1, 1, 0, -1, 1, 1, -1, 0, 0, -2, 1, 2, 0, 2, -1, -5, -1, 1, 3, 7, 1, -8, -8, -6, 3, 18, 16, 0, -17, -31, -21, 19, 51, 47, 3, -70, -106, -48, 71, 170, 156, -18, -243, -318, -132, 253, 564, 455, -130, -819, -1024, -341, 952, 1849, 1355, -606, -2789, -3199, -727, 3410, 5979, 3932, -2678, -9408, -9926, -1281, 12047

Offset: 0

N. J. A. Sloane, Jun 01 2009

sign

This g.f. multiplied by h(k) for k large (cf. A151552) gives the g.f. for (A160573 prefixed by an initial 1).

Cf. A151676, A160573, A151552.

A151677 a(n) = sum_{k >= 0} binomial(3*wt(n+k),k), where wt() = A000120().

Original entry on oeis.org

2, 3, 4, 6, 11, 3, 6, 10, 11, 19, 6, 6, 11, 8, 14, 15, 11, 19, 15, 26, 36, 3, 6, 10, 11, 19, 8, 12, 21, 19, 33, 21, 11, 19, 15, 26, 36, 19, 33, 32, 49, 76, 8, 6, 11, 8, 14, 15, 11, 19, 15, 26, 36, 8, 14, 17, 23, 40, 16, 26, 38, 42, 73, 28, 11, 19, 15, 26, 36, 19, 33, 32, 49, 76, 17, 26, 38

Offset: 0

N. J. A. Sloane, Jun 01 2009

nonn

Suggested by Hagen von Eitzen's formula for A160573.

cp's OEIS Frontend

A151687 G.f.: x + x^2 * Product_{n>=0} (1 + x^(2^n-1) + x^(2^n)).

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A151683 Irregular triangle read by rows: row n (n>=0) gives binomial(wt(n+k),k), k >= 0, up to the point where the terms are all zeros (wt() = A000120()).

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A151684 Let c(n) = x^(2^n-1)(1-x^(2^n)), g(n) = 1 + x^(2^n-1) + x^(2^n), h(n) = Product_{i=1..n} g(i); then use g.f. (1+2x) - Sum_{n>=1} c(n)/h(n).

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Author

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Crossrefs

A151677 a(n) = sum_{k >= 0} binomial(3*wt(n+k),k), where wt() = A000120().

Views

Author

Keywords

Comments

cp's OEIS Frontend

A151687 G.f.: x + x^2 * Product_{n>=0} (1 + x^(2^n-1) + x^(2^n)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

A151683 Irregular triangle read by rows: row n (n>=0) gives binomial(wt(n+k),k), k >= 0, up to the point where the terms are all zeros (wt() = A000120()).

Views

Author

Keywords

Comments

Examples

Crossrefs

A151684 Let c(n) = x^(2^n-1)*(1-x^(2^n)), g(n) = 1 + x^(2^n-1) + x^(2^n), h(n) = Product_{i=1..n} g(i); then use g.f. (1+2*x) - Sum_{n>=1} c(n)/h(n).

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Author

Keywords

Comments

Crossrefs

A151677 a(n) = sum_{k >= 0} binomial(3*wt(n+k),k), where wt() = A000120().

Views

Author

Keywords

Comments

A151684 Let c(n) = x^(2^n-1)(1-x^(2^n)), g(n) = 1 + x^(2^n-1) + x^(2^n), h(n) = Product_{i=1..n} g(i); then use g.f. (1+2x) - Sum_{n>=1} c(n)/h(n).