A306847 -id:A306847 - cp's OEIS frontend

A306846 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^k).

1, 1, 2, 1, 1, 4, 1, 1, 2, 8, 1, 1, 1, 4, 16, 1, 1, 1, 2, 8, 32, 1, 1, 1, 1, 5, 16, 64, 1, 1, 1, 1, 2, 11, 32, 128, 1, 1, 1, 1, 1, 6, 22, 64, 256, 1, 1, 1, 1, 1, 2, 16, 43, 128, 512, 1, 1, 1, 1, 1, 1, 7, 36, 85, 256, 1024, 1, 1, 1, 1, 1, 1, 2, 22, 72, 170, 512, 2048

Offset: 0

Seiichi Manyama, Mar 13 2019

			Square array begins:
     1,   1,  1,  1,  1,  1, 1, 1, 1, ...
     2,   1,  1,  1,  1,  1, 1, 1, 1, ...
     4,   2,  1,  1,  1,  1, 1, 1, 1, ...
     8,   4,  2,  1,  1,  1, 1, 1, 1, ...
    16,   8,  5,  2,  1,  1, 1, 1, 1, ...
    32,  16, 11,  6,  2,  1, 1, 1, 1, ...
    64,  32, 22, 16,  7,  2, 1, 1, 1, ...
   128,  64, 43, 36, 22,  8, 2, 1, 1, ...
   256, 128, 85, 72, 57, 29, 9, 2, 1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-9 give A000079, A011782, A024493, A038503, A139398, A306847, A306852, A306859, A306860.

Cf. A306680.

T[n_, k_] := Sum[Binomial[n, k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n,k*j).

A119336 Expansion of (1-x)^4/((1-x)^6 - x^6).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 16, 45, 130, 341, 804, 1730, 3460, 6555, 12016, 21845, 40410, 77540, 155080, 320001, 669526, 1398101, 2884776, 5858126, 11716252, 23166783, 45536404, 89478485, 176565486, 350739488, 701478976, 1410132405, 2841788170

Offset: 0

Paul Barry, May 14 2006

Row sums of A119335. Binomial transform of (1+x)/(1-x)^6.

Equals binomial transform of [1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, ...]. - Gary W. Adamson, Mar 14 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Cf. A119335, A306847.

CoefficientList[Series[(1-x)^4/((1-x)^6-x^6),{x,0,40}],x] (* or *) LinearRecurrence[{6,-15,20,-15,6},{1,2,3,4,5},40] (* Harvey P. Dale, Dec 25 2015 *)

{a(n) = sum(k=0, n\6, binomial(n+1, 6*k+1))} \\ Seiichi Manyama, Mar 22 2019

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5), with a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=5. - Harvey P. Dale, Dec 25 2015
a(n) = Sum_{k=0..floor(n/6)} binomial(n+1,6*k+1). - Seiichi Manyama, Mar 22 2019

A307040 a(n) = Sum_{k=0..floor(n/6)} (-1)^kbinomial(n,6k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, -6, -27, -83, -209, -461, -922, -1702, -2911, -4549, -6187, -6187, 0, 23238, 87021, 238337, 564719, 1217483, 2434966, 4542526, 7865521, 12403951, 16942381, 16942381, 0, -63239286, -236031147, -644886923, -1525870529, -3287837741, -6575675482

Offset: 0

Seiichi Manyama, Mar 21 2019

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-2).

Column 6 of A307039.

Cf. A306847.

a[n_] := Sum[(-1)^k * Binomial[n,6*k], {k,0,Floor[n/6]}]; Array[a, 37, 0] (* Amiram Eldar, May 25 2021 *)

{a(n) = sum(k=0, n\6, (-1)^k*binomial(n, 6*k))}

N=66; x='x+O('x^N); Vec((1-x)^5/((1-x)^6+x^6))

G.f.: (1 - x)^5/((1 - x)^6 + x^6).

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - 2*a(n-6) for n > 5.

cp's OEIS Frontend

A306846 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^k).

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A119336 Expansion of (1-x)^4/((1-x)^6 - x^6).

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A307040 a(n) = Sum_{k=0..floor(n/6)} (-1)^kbinomial(n,6k).

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cp's OEIS Frontend

A306846 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^k).

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Mathematica

Formula

A119336 Expansion of (1-x)^4/((1-x)^6 - x^6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A307040 a(n) = Sum_{k=0..floor(n/6)} (-1)^k*binomial(n,6*k).

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Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A307040 a(n) = Sum_{k=0..floor(n/6)} (-1)^kbinomial(n,6k).