cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A362179 Main diagonal of the square array A058395.

Original entry on oeis.org

1, 1, 4, 10, 25, 61, 146, 344, 800, 1840, 4192, 9472, 21248, 47360, 104960, 231424, 507904, 1110016, 2416640, 5242880, 11337728, 24444928, 52559872, 112721920, 241172480, 514850816, 1096810496, 2332033024, 4949278720, 10485760000, 22179479552, 46841987072
Offset: 0

Views

Author

Philippe Deléham, Apr 10 2023

Keywords

Comments

Binomial transform of 1, 0, 3, 0, 6, 0, 10, 0, 15, 0, ... (triangular numbers alternating with zeros).

Links

Crossrefs

Cf. A000217, A007318, A008805, A058395, A158920.

Formula

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n > 5.
a(n) = A158920(n+1) - A158920(n).
a(n) = Sum_{k=0..n} A007318(n,k)*A008805(k)*(1 + (-1)^k)/2.
G.f.: (1-x)^5/(1-2*x)^3.

A058396 Expansion of ((1-x)/(1-2*x))^3.

Original entry on oeis.org

1, 3, 9, 25, 66, 168, 416, 1008, 2400, 5632, 13056, 29952, 68096, 153600, 344064, 765952, 1695744, 3735552, 8192000, 17891328, 38928384, 84410368, 182452224, 393216000, 845152256, 1811939328, 3875536896, 8271167488, 17616076800
Offset: 0

Views

Author

Henry Bottomley, Nov 24 2000

Keywords

Comments

If X_1,X_2,...,X_n are 2-blocks of a (2n+3)-set X then, for n>=1, a(n+1) is the number of (n+2)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
Equals row sums of triangle A152230. - Gary W. Adamson, Nov 29 2008
a(n) is the number of weak compositions of n with exactly 2 parts equal to 0. - Milan Janjic, Jun 27 2010
Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^3; see A291000. - Clark Kimberling, Aug 24 2017

Links

Crossrefs

Cf. A045623, A001793, A152230. A diagonal of A058395.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^3)); // G. C. Greubel, Oct 16 2018
  • Maple
    seq(coeff(series(((1-x)/(1-2*x))^3,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 16 2018
  • Mathematica
    CoefficientList[ Series[(1 - x)^3/(1 - 2x)^3, {x, 0, 28}], x] (* Robert G. Wilson v, Jun 28 2005 *)
Join[{1},LinearRecurrence[{6,-12,8},{3,9,25},40]] (* Harvey P. Dale, Oct 17 2011 *)
  • PARI
    Vec((1-x)^3/(1-2*x)^3+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Formula

a(n) = (n+2)*(n+7)*2^(n-4) for n > 0.
a(n) = Sum_{k=0..floor((n+2)/2)} C(n+2, 2k)*k(k+1)/2. - Paul Barry, May 15 2003
Binomial transform of quarter squares A002620 (without leading zeros). - Paul Barry, May 27 2003
a(n) = Sum_{k=0..n} C(n, k)*floor((k+2)^2/4). - Paul Barry, May 27 2003
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), n > 3. - Harvey P. Dale, Oct 17 2011
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 145189/525 - 1984*log(2)/5.
Sum_{n>=0} (-1)^n/a(n) = 30103/175 - 2112*log(3/2)/5. (End)
E.g.f.: (1 + exp(2*x)*(7 + 10*x + 2*x^2))/8. - Stefano Spezia, Feb 01 2025

A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 3, 1, 0, 1, 2, 4, 4, 1, 1, 1, 2, 4, 7, 5, 1, 0, 1, 2, 4, 8, 11, 6, 1, 1, 1, 2, 4, 8, 15, 16, 7, 1, 0, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 0, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 0
Offset: 0

Views

Author

Henry Bottomley, Nov 24 2000

Keywords

Comments

Changing the formula by replacing T(0,2n)=T(1,n) by T(0,2n)=T(m,n) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058394, A058395, A057884, (and effectively A007318).

Examples

			Rows are (1,0,1,0,1,0,1,...), (1,1,1,1,1,1,...), (1,2,2,2,2,2,...), (1,3,4,4,4,...) etc.

Crossrefs

Rows are A000035 (A000012 with zeros), A000012, A040000 etc. Columns are A000012, A001477, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863 etc. Diagonals include A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, etc. The triangles A008949, A054143 and A055248 also appear in the half of the array which is not powers of 2.

Formula

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(1, 1)=1, T(0, 2n)=T(1, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2).

A058394 A square array based on natural numbers (A000027) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 2, 2, 1, 3, 2, 3, 3, 1, 0, 3, 4, 5, 4, 1, 4, 3, 5, 7, 8, 5, 1, 0, 4, 6, 9, 12, 12, 6, 1, 5, 4, 7, 11, 16, 20, 17, 7, 1, 0, 5, 8, 13, 20, 28, 32, 23, 8, 1, 6, 5, 9, 15, 24, 36, 48, 49, 30, 9, 1, 0, 6, 10, 17, 28, 44, 64, 80, 72, 38, 10, 1, 7, 6, 11, 19, 32, 52, 80, 112, 129
Offset: 0

Views

Author

Henry Bottomley, Nov 24 2000

Keywords

Comments

Changing the formula by replacing T(2n,0)=T(n,2) by T(2n,0)=T(n,m) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058393, A058395, A057884 (and effectively A007318).

Examples

			Rows are (1,0,2,0,3,0,4,...), (1,1,2,2,3,3,...), (1,2,3,4,5,6,...), (1,3,5,7,9,11,...), etc.

Crossrefs

Rows are A027656 (A000027 with zeros), A008619, A000027, A005408, A008574 etc. Columns are A000012, A001477, A022856 etc. Diagonals include A034007, A045891, A045623, A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 etc. The triangle A055249 also appears in half of the array.

Formula

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(2n, 0)=T(n, 2) and T(2n+1, 0)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2)^2.
Showing 1-4 of 4 results.

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