A306680 -id:A306680 - cp's OEIS frontend

A005676 a(n) = Sum_{k=0..n} C(n-k,4*k).

1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 128, 220, 376, 661, 1211, 2290, 4382, 8347, 15706, 29191, 53824, 99009, 182497, 337745, 627401, 1167937, 2174834, 4046070, 7517368, 13951852, 25880583, 48009456, 89090436, 165392856, 307137901

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,1,4).
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1).

Column k=4 of A306680.

[&+[Binomial(n-k, 4*k): k in [0..n]]: n in [0..40]]; // Vincenzo Librandi, Sep 08 2017

A005676:=(z-1)**3/(-1+4*z-6*z**2+4*z**3-z**4+z**5); # Simon Plouffe in his 1992 dissertation.

LinearRecurrence[{4, -6, 4, -1, 1}, {1, 1, 1, 1, 1}, 40] (* or *) CoefficientList[Series[(1 - x)^3 / ((1 - x)^4 - x^5), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 08 2017 *)

From Paul Barry, Jul 23 2004: (Start)
G.f.: (1-3x+3x^2-x^3)/(1-4x+6x^2-4x^3+x^4-x^5) = (1-x)^3/((1-x)^4-x^5).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, 4k).
a(n) = 4a(n-1)-6a(n-2)+4a(n-3)-a(n-4)+a(n-5). (End)

More terms from James Sellers, Aug 21 2000

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).

Original entry on oeis.org

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).

A306713 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5, 1, 0, 0, 1, 1, 8, 1, 0, 0, 0, 1, 2, 13, 1, 0, 0, 0, 1, 0, 2, 21, 1, 0, 0, 0, 0, 1, 1, 3, 34, 1, 0, 0, 0, 0, 1, 0, 2, 4, 55, 1, 0, 0, 0, 0, 0, 1, 0, 1, 5, 89, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 7, 144, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 9, 233

Offset: 0

Seiichi Manyama, Mar 05 2019

A(n,k) is the number of compositions of n into parts k and k+1.

			Square array begins:
    1, 1, 1, 1, 1, 1, 1, 1, 1, ...
    1, 0, 0, 0, 0, 0, 0, 0, 0, ...
    2, 1, 0, 0, 0, 0, 0, 0, 0, ...
    3, 1, 1, 0, 0, 0, 0, 0, 0, ...
    5, 1, 1, 1, 0, 0, 0, 0, 0, ...
    8, 2, 0, 1, 1, 0, 0, 0, 0, ...
   13, 2, 1, 0, 1, 1, 0, 0, 0, ...
   21, 3, 2, 0, 0, 1, 1, 0, 0, ...
   34, 4, 1, 1, 0, 0, 1, 1, 0, ...
   55, 5, 1, 2, 0, 0, 0, 1, 1, ...

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

Columns 1-10 give A000045(n+1), A182097, A017817, A017827, A017837, A017847, A017857, A017867, A017877, A017887.

Cf. A306646, A306680.

T[n_, k_] := Sum[Binomial[j, n-k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

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

A306735 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. ((k+1-x)*(1-x)^(k-1))/((1-x)^k-x^(k+1)).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 3, 1, 5, 3, 2, 4, 1, 6, 4, 3, 5, 7, 1, 7, 5, 4, 3, 10, 11, 1, 8, 6, 5, 4, 7, 17, 18, 1, 9, 7, 6, 5, 4, 18, 29, 29, 1, 10, 8, 7, 6, 5, 9, 39, 51, 47, 1, 11, 9, 8, 7, 6, 5, 28, 73, 90, 76, 1, 12, 10, 9, 8, 7, 6, 11, 74, 127, 158, 123, 1, 13, 11, 10, 9, 8, 7, 6, 40, 164, 219, 277, 199, 1

Offset: 0

Seiichi Manyama, Mar 06 2019

			Square array begins:
   1,  2,  3,   4,   5,   6,  7,  8, 9, ...
   1,  1,  2,   3,   4,   5,  6,  7, 8, ...
   1,  3,  2,   3,   4,   5,  6,  7, 8, ...
   1,  4,  5,   3,   4,   5,  6,  7, 8, ...
   1,  7, 10,   7,   4,   5,  6,  7, 8, ...
   1, 11, 17,  18,   9,   5,  6,  7, 8, ...
   1, 18, 29,  39,  28,  11,  6,  7, 8, ...
   1, 29, 51,  73,  74,  40, 13,  7, 8, ...
   1, 47, 90, 127, 164, 125, 54, 15, 8, ...

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

Columns 0-2 give A000012, A000032, A259967.

Cf. A306646, A306680.

A(n,k) = A306646(k*n,k) for k > 0.

A(n,k) = (k+1)*A306680(n,k) - A306680(n-1,k) for n > 0.

A306721 a(n) = Sum_{k=0..n} binomial(k, 7*(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1717, 3433, 6437, 11456, 19569, 32505, 53449, 89149, 155041, 286825, 564929, 1163317, 2442210, 5117225, 10558381, 21308121, 41973391, 80778601, 152344397, 282855561, 520060249, 953217792, 1753553441, 3256528177, 6127896977, 11694334137

Offset: 0

Seiichi Manyama, Mar 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..3556
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1,1).

Column 7 of A306680.

Cf. A017857.

a[n_] := Sum[Binomial[k, 7*(n-k)], {k, 0, n}]; Array[a, 40, 0] (* Amiram Eldar, Jun 21 2021 *)

{a(n) = sum(k=0, n, binomial(k, 7*(n-k)))}

N=66; x='x+O('x^N); Vec((1-x)^6/((1-x)^7-x^8))

G.f.: (1-x)^6/((1-x)^7-x^8).
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7)+a(n-8) for n > 7.
a(n) = A017857(7*n).

A306752 a(n) = Sum_{k=0..n} binomial(k, 8*(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 10, 46, 166, 496, 1288, 3004, 6436, 12871, 24312, 43776, 75736, 126940, 208336, 340120, 564928, 980629, 1817047, 3605252, 7531836, 16146326, 34716826, 73737316, 153430156, 311652271, 617594122, 1195477615, 2266064352, 4221317464

Offset: 0

Seiichi Manyama, Mar 07 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3528
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1,1).

Column 8 of A306680.

Cf. A017867.

a[n_] := Sum[Binomial[k, 8*(n-k)], {k, 0, n}]; Array[a, 38, 0] (* Amiram Eldar, Jun 21 2021 *)

{a(n) = sum(k=0, n, binomial(k, 8*(n-k)))}

N=66; x='x+O('x^N); Vec((1-x)^7/((1-x)^8-x^9))

G.f.: (1-x)^7/((1-x)^8 - x^9).
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + a(n-9) for n > 8.
a(n) = A017867(8*n).

A306753 a(n) = Sum_{k=0..n} binomial(k, 9*(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 11, 56, 221, 716, 2003, 5006, 11441, 24311, 48621, 92380, 167980, 294121, 498751, 824506, 1341154, 2177572, 3605251, 6249101, 11593726, 23138117, 48904469, 106653707, 234305936, 510034166, 1089810953, 2275676459, 4637090547

Offset: 0

Seiichi Manyama, Mar 07 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3506
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1,1).

Column 9 of A306680.

Cf. A017877.

a[n_] := Sum[Binomial[k, 9*(n-k)], {k, 0, n}]; Array[a, 38, 0] (* Amiram Eldar, Jun 21 2021 *)

{a(n) = sum(k=0, n, binomial(k, 9*(n-k)))}

N=66; x='x+O('x^N); Vec((1-x)^8/((1-x)^9-x^10))

G.f.: (1-x)^8/((1-x)^9 - x^10).
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) + a(n-10) for n > 9.
a(n) = A017877(9*n).

A348315 a(n) = Sum_{k=0..n} binomial(n^2 - k,n*k).

Original entry on oeis.org

1, 1, 4, 64, 4382, 1357136, 1597653852, 8389021518585, 164828345435877580, 14256525628649472111712, 4602970880920727147946847283, 6484132480933772335644792339409450, 34112054985056318746734374876035089268523

Offset: 0

Seiichi Manyama, Oct 11 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..57

Cf. A167009, A238696, A306680, A348322.

a[n_] := Sum[Binomial[n^2 - k, n*k], {k, 0, n}]; Array[a, 13, 0] (* Amiram Eldar, Oct 12 2021 *)

a(n) = sum(k=0, n, binomial(n^2-k, n*k));

a(n) = polcoef((1-x)^(n-1)/((1-x)^n-x^(n+1)+x*O(x^n^2)), n^2);

a(n) = A306680(n^2,n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^(n+1)).

a(n) ~ c * 2^(1/2 - n/2 + n^2) / (sqrt(Pi)*exp(1/8)*n), where c = Sum_{m = -oo..+oo} 1/(2^m * exp(m*(2*m+1))) = 1.77058122254033174512511... if n is even and c = Sum_{m = -oo..+oo} 1/(2^(m + 1/2) * exp((m+1)*(2*m+1))) = 1.81629595919505881855931... if n is odd. - Vaclav Kotesovec, Oct 12 2021

cp's OEIS Frontend

A005676 a(n) = Sum_{k=0..n} C(n-k,4*k).

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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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A306713 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

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A306735 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. ((k+1-x)*(1-x)^(k-1))/((1-x)^k-x^(k+1)).

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A306721 a(n) = Sum_{k=0..n} binomial(k, 7*(n-k)).

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A306752 a(n) = Sum_{k=0..n} binomial(k, 8*(n-k)).

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A306753 a(n) = Sum_{k=0..n} binomial(k, 9*(n-k)).

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Mathematica

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A348315 a(n) = Sum_{k=0..n} binomial(n^2 - k,n*k).

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