A003522 -id:A003522 - cp's OEIS frontend

A306680 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^(k+1)).

1, 1, 2, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 3, 5, 1, 1, 1, 2, 5, 6, 1, 1, 1, 1, 4, 8, 7, 1, 1, 1, 1, 2, 7, 13, 8, 1, 1, 1, 1, 1, 5, 12, 21, 9, 1, 1, 1, 1, 1, 2, 11, 21, 34, 10, 1, 1, 1, 1, 1, 1, 6, 21, 37, 55, 11, 1, 1, 1, 1, 1, 1, 2, 16, 37, 65, 89, 12

Offset: 0

Seiichi Manyama, Mar 05 2019

			A(4,1) = A306713(4,1) = 5, A(4,2) = A306713(8,2) = 4.
Square array begins:
   1,  1,  1,  1,  1,  1, 1, 1, 1, ...
   2,  1,  1,  1,  1,  1, 1, 1, 1, ...
   3,  2,  1,  1,  1,  1, 1, 1, 1, ...
   4,  3,  2,  1,  1,  1, 1, 1, 1, ...
   5,  5,  4,  2,  1,  1, 1, 1, 1, ...
   6,  8,  7,  5,  2,  1, 1, 1, 1, ...
   7, 13, 12, 11,  6,  2, 1, 1, 1, ...
   8, 21, 21, 21, 16,  7, 2, 1, 1, ...
   9, 34, 37, 37, 36, 22, 8, 2, 1, ...

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

Columns 0-9 give A000027(n+1), A000045(n+1), A005251(n+1), A003522, A005676, A099132, A293169, A306721, A306752, A306753.

Cf. A306646, A306713, A306735.

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

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

A(n,k) = A306713(k*n,k) for k > 0.

A099785 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3k) 2^(n-3*k).

Original entry on oeis.org

1, 2, 4, 8, 18, 48, 144, 448, 1380, 4152, 12224, 35456, 102024, 292768, 840416, 2416384, 6959504, 20069280, 57913536, 167158656, 482462752, 1392319488, 4017460224, 11590946816, 33439639616, 96470796672, 278311599616

Offset: 0

Paul Barry, Oct 26 2004

In general a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * u^k * v^(n-4*k) has g.f. (1-v*x)^2/((1-v*x)^3 - u*x^4) and satisfies the recurrence a(n) = 3*v*a(n-1) - 3*v^2*a(n-2) + v^3*a(n-3) + u*a(n-4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8,2).

Cf. A003522, A097119.

Cf. A099780, A099781, A099783, A099784, A099785, A099786, A099787.

a:=[1,2,4,8];; for n in [5..30] do a[n]:=6*a[n-1] -12*a[n-2] + 8*a[n-3] +2*a[n-4]; od; a; # G. C. Greubel, Sep 04 2019

I:=[1,2,4,8]; [n le 4 select I[n] else 6*Self(n-1) - 12*Self(n-2) + 8*Self(n-3) + 2*Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 04 2019

seq(coeff(series((1-2*x)^2/((1-2*x)^3 - 2*x^4), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Sep 04 2019

Table[Sum[Binomial[n-k,3k]2^(n-3k),{k,0,Floor[n/4]}],{n,0,30}] (* or *) LinearRecurrence[{6,-12,8,2},{1,2,4,8},30] (* Harvey P. Dale, Apr 01 2012 *)

my(x='x+O('x^30)); Vec((1-2*x)^2/((1-2*x)^3 - 2*x^4)) \\ G. C. Greubel, Sep 04 2019

def A099785_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-2*x)^2/((1-2*x)^3 - 2*x^4)).list()
A099785_list(30) # G. C. Greubel, Sep 04 2019

G.f.: (1-2*x)^2/((1-2*x)^3 - 2*x^4).

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) + 2*a(n-4).

A099786 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3k)3^(n-4*k).

Original entry on oeis.org

1, 3, 9, 27, 82, 255, 819, 2727, 9397, 33312, 120537, 441855, 1631017, 6036879, 22345074, 82589247, 304612975, 1120960983, 4116353265, 15088372416, 55224373105, 201895801851, 737506551321, 2692518758163, 9826402960882

Offset: 0

Paul Barry, Oct 26 2004

In general a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * u^k * v^(n-4*k) has g.f. (1-v*x)^2/((1-v*x)^3 - u*x^4) and satisfies the recurrence a(n) = 3*v*a(n-1) - 3*v^2*a(n-2) + v^3*a(n-3) + u*a(n-4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27,1).

Cf. A003522, A097119.

Cf. A099780, A099781, A099782, A099783, A099784, A099785, A099787.

a:=[1,3,9,27];; for n in [5..30] do a[n]:=9*a[n-1]-27*a[n-2] + 27*a[n-3] +a[n-4]; od; a; # G. C. Greubel, Sep 04 2019

I:=[1,3,9,27]; [n le 4 select I[n] else 9*Self(n-1) - 27*Self(n-2) + 27*Self(n-3) +Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 04 2019

seq(coeff(series((1-3*x)^2/((1-3*x)^3 - x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019

LinearRecurrence[{9,-27,27,1},{1,3,9,27},40] (* or *) CoefficientList[ Series[-((1-3 x)^2/(x (x (x (x+27)-27)+9)-1)),{x,0,40}],x] (* Harvey P. Dale, Jun 06 2011 *)

my(x='x+O('x^30)); Vec((1-3*x)^2/((1-3*x)^3 - x^4)) \\ G. C. Greubel, Sep 04 2019

def A099786_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)^2/((1-3*x)^3 - x^4)).list()
A099786_list(30) # G. C. Greubel, Sep 04 2019

G.f.: (1-3*x)^2/((1-3*x)^3 - x^4).

a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) + a(n-4).

A373906 a(n) = Sum_{k=0..floor(n/4)} binomial(n+2k,n-4k).

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 29, 85, 212, 476, 1016, 2172, 4825, 11213, 26763, 64095, 151851, 354737, 820328, 1889968, 4361521, 10106859, 23509678, 54793282, 127709888, 297336790, 691382201, 1606284377, 3731020629, 8668253125, 20146856893, 46840732201, 108918637566, 253262275888

Offset: 0

Seiichi Manyama, Jun 22 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-14,6,-1).

Cf. A003269, A003522, A005252, A038503, A099131.

Cf. A107025, A373904, A373905.

a(n) = sum(k=0, n\4, binomial(n+2*k, n-4*k));

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 14*a(n-4) + 6*a(n-5) - a(n-6).

G.f.: 1/(1 - x - x^4/(1 - x)^5).

A096750 Expansion of (1-x+x^2)/(1-2x+2x^2-x^3-x^4).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 6, 7, 8, 12, 21, 33, 44, 55, 76, 119, 185, 263, 351, 480, 706, 1066, 1551, 2156, 2982, 4269, 6281, 9162, 13013, 18252, 25921, 37513, 54449, 78045, 110626, 157124, 225490, 325403, 467576, 666960, 949661, 1358381, 1951976, 2803811

Offset: 0

Paul Barry, Jul 07 2004

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,1,1).

Cf. A003522.

CoefficientList[Series[(1-x+x^2)/(1-2x+2x^2-x^3-x^4),{x,0,50}],x] (* or *) LinearRecurrence[ {2,-2,1,1},{1,1,1,1},50] (* Harvey P. Dale, Jun 16 2024 *)

a(n) = 2*a(n-1)-2*a(n-2)+a(n-3)+a(n-4).

A138653 a(n) = 3a(n-1)-3a(n-2)+a(n-3)+a(n-4).

Original entry on oeis.org

1, 2, 4, 8, 15, 27, 48, 86, 156, 285, 521, 950, 1728, 3140, 5707, 10379, 18884, 34362, 62520, 113737, 206897, 376362, 684652, 1245504, 2265815, 4121947, 7498552, 13641134, 24815508, 45143621, 82124025, 149397854, 271780616, 494415932, 899427827, 1636214155

Offset: 0

Paul Curtz, May 15 2008

Sequence is identical to its shifted third differences.

Colin Barker, Table of n, a(n) for n = 0..1000
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1).

Cf. A098057, A003522.

LinearRecurrence[{3, -3, 1, 1},{1, 2, 4, 8},14] (* Ray Chandler, Sep 23 2015 *)

Vec((1-x+x^2+x^3)/(1-3*x+3*x^2-x^3-x^4) + O(x^50)) \\ Colin Barker, Oct 18 2016

G.f.: (1-x+x^2+x^3) / (1-3*x+3*x^2-x^3-x^4). - Colin Barker, Oct 18 2016

A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4k,3k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 57, 85, 122, 173, 249, 371, 575, 918, 1485, 2398, 3830, 6030, 9369, 14422, 22107, 33909, 52226, 80888, 125925, 196706, 307653, 480873, 750275, 1168085, 1815191, 2817518, 4371772, 6785606, 10539893, 16384908, 25488736

Offset: 0

Seiichi Manyama, Feb 28 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,1).

Cf. A003522, A100134, A137356.

Cf. A003520, A005689, A348289.

LinearRecurrence[{3, -3, 1, 0, 0, 0, 1}, Table[1, 7], 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\7, binomial(n-4*k, 3*k));

my(N=50, x='x+O('x^N)); Vec((1-x)^2/((1-x)^3-x^7))

G.f.: (1-x)^2/((1-x)^3 - x^7).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-7).

A375169 Expansion of (1 - x) / ((1 - x)^3 - x^4).

Original entry on oeis.org

1, 2, 3, 4, 6, 11, 22, 43, 80, 144, 257, 462, 839, 1532, 2798, 5099, 9274, 16855, 30640, 55728, 101393, 184490, 335659, 610628, 1110790, 2020635, 3675822, 6686979, 12164896, 22130208, 40258737, 73237462, 133231279, 242370396, 440913550, 802098203, 1459155634

Offset: 0

Seiichi Manyama, Aug 05 2024

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1).

Cf. A024494, A052921, A124820, A137357.

Cf. A003522, A107068.

my(N=40, x='x+O('x^N)); Vec((1-x)/((1-x)^3-x^4))

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+1-k,n-4*k).
a(n) = (n + 1)*hypergeom([(1-n)/4, (2-n)/4, (3-n)/4, -n/4], [2/3, 4/3, -1-n], -4^4/3^3). - Stefano Spezia, Jun 18 2025

cp's OEIS Frontend

A306680 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^(k+1)).

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A099785 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * 2^(n-3*k).

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A099786 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k)*3^(n-4*k).

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A373906 a(n) = Sum_{k=0..floor(n/4)} binomial(n+2*k,n-4*k).

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A096750 Expansion of (1-x+x^2)/(1-2x+2x^2-x^3-x^4).

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A138653 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+a(n-4).

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A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4*k,3*k).

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A099785 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3k) 2^(n-3*k).

A099786 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3k)3^(n-4*k).

A373906 a(n) = Sum_{k=0..floor(n/4)} binomial(n+2k,n-4k).

A138653 a(n) = 3a(n-1)-3a(n-2)+a(n-3)+a(n-4).

A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4k,3k).