A115292 -id:A115292 - cp's OEIS frontend

A115291 Expansion of (1+x)^3/(1-x).

1, 4, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Paul Barry, Jan 19 2006

Partial sums are A086570. Partial sums of squares are A115295. Correlation triangle is A115292.
Let m=4. We observe that a(n) = Sum_{k=0..floor(n/2)} C(m,n-2*k). Then there is a link with A113311 and A040000: it is the same formula with respectively m=3 and m=2. We can generalize this result with the sequence whose G.f is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009
Also continued fraction expansion of (132-sqrt(17))/103. - Bruno Berselli, Sep 23 2011
Also decimal expansion of 1331/9000. - Vincenzo Librandi, Sep 23 2011

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

Cf. A040000, A113311, A171418, A171440, A171441, A171442, A171443.

CoefficientList[Series[(1+x)^3/(1-x),{x,0,100}],x] (* or *) PadRight[ {1,4,7},120,{8}] (* Harvey P. Dale, May 23 2016 *)

a(n) = 8 - C(2, n) - 2*C(1, n) - 4*C(0, n).
a(n) = Sum_{k=0..n} C(3, k).
a(n) = A004070(n, 3).
From Elmo R. Oliveira, Aug 09 2024: (Start)
E.g.f.: 8*exp(x) - 7 - 4*x - x^2/2.
a(n) = 8, n > 2. (End)

A115293 Row sums of correlation triangle for (1+x)^3/(1-x).

Original entry on oeis.org

1, 8, 31, 80, 160, 272, 416, 592, 800, 1040, 1312, 1616, 1952, 2320, 2720, 3152, 3616, 4112, 4640, 5200, 5792, 6416, 7072, 7760, 8480, 9232, 10016, 10832, 11680, 12560, 13472, 14416, 15392, 16400, 17440, 18512, 19616, 20752, 21920, 23120, 24352

Offset: 0

Paul Barry, Jan 19 2006

Row sums of number triangle A115292.

If Y_i (i=1,2,3,4,5) are 2-blocks of a (n+5)-set X then a(n-2) is the number of 7-subsets of X intersecting each Y_i (i=1,2,3,4,5). - Milan Janjic, Oct 28 2007

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A115291, A115292, A262732.

seq(add(binomial(5,n-k)*binomial(k+2,k), k = 0..n), n = 0..40); # Peter Bala, Sep 26 2021

LinearRecurrence[{3,-3,1},{1,8,31,80,160,272},50] (* Harvey P. Dale, Dec 03 2018 *)

a(n) = sum(k = 0, n, binomial(5,n-k)*binomial(k+2,k)); \\ Michel Marcus, Oct 01 2021

G.f.: A(x) = (1+x)^5/(1-x)^3.
a(n) = Sum_{k = 0..n} Sum_{j = 0..n} [j<=k]*A115291(k-j)*[j<=n-k]*A115291(n-k-j).
From Peter Bala, Sep 26 2021: (Start)
a(n) = Sum_{k = 0..n} binomial(5,n-k)*binomial(k+2,k).
A262732(n) = [x^n] A(x)^n. (End)

A115295 Partial sums of squares of A115291(n).

Original entry on oeis.org

1, 17, 66, 130, 194, 258, 322, 386, 450, 514, 578, 642, 706, 770, 834, 898, 962, 1026, 1090, 1154, 1218, 1282, 1346, 1410, 1474, 1538, 1602, 1666, 1730, 1794, 1858, 1922, 1986, 2050, 2114, 2178, 2242, 2306, 2370, 2434, 2498, 2562, 2626, 2690, 2754, 2818

Offset: 0

Paul Barry, Jan 19 2006

Central coefficients of number triangle A115292.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Join[{1, 17}, Range[66, 7000, 64]] (* Vladimir Joseph Stephan Orlovsky, Jul 12 2011 *)

a(n)=sum{k=0..n, (sum{j=0..k, C(3, j)})^2}; a(n)=A115292(2n, n).

A115294 Diagonal sums of correlation triangle for (1+x)^3/(1-x).

Original entry on oeis.org

1, 4, 11, 25, 47, 79, 122, 175, 239, 314, 399, 495, 602, 719, 847, 986, 1135, 1295, 1466, 1647, 1839, 2042, 2255, 2479, 2714, 2959, 3215, 3482, 3759, 4047, 4346, 4655, 4975, 5306, 5647, 5999, 6362, 6735, 7119, 7514, 7919, 8335, 8762, 9199, 9647, 10106

Offset: 0

Paul Barry, Jan 19 2006

Diagonal sums of number triangle A115292.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

LinearRecurrence[{2,-1,1,-2,1},{1,4,11,25,47,79,122,175,239},50] (* Harvey P. Dale, Jun 11 2017 *)

G.f.: (1+x)^2*(1+x^2)^3/((1-x)^2*(1-x^3)); a(n)=sum{k=0..floor(n/2), sum{j=0..n-k, [j<=k]*A115291(k-j)*[j<=n-2k]*A115291(n-2k-j)}}.

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A115291 Expansion of (1+x)^3/(1-x).

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A115293 Row sums of correlation triangle for (1+x)^3/(1-x).

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A115295 Partial sums of squares of A115291(n).

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A115294 Diagonal sums of correlation triangle for (1+x)^3/(1-x).

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