A113310 -id:A113310 - cp's OEIS frontend

A113311 Expansion of (1+x)^2/(1-x).

1, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Paul Barry, Oct 25 2005

Row sums of A113310.
Let m=3. We observe that a(n)=Sum_{k=0..floor(n/2)} C(m,n-2*k). Then there is a link with A040000 and A115291: it is the same formula with respectively m=2 and m=4. 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 (3+sqrt(5))/4. - Bruno Berselli, Sep 23 2011
Also decimal expansion of 121/900. - Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A115291, A171418, A171440-A171443.

CoefficientList[Series[(1+x)^2/(1-x),{x,0,110}],x] (* Harvey P. Dale, Aug 19 2011 *)

a(n)=if(n>1,4,2*n+1) \\ Charles R Greathouse IV, Jun 12 2015

a(n) = Sum_{k=0..n} Sum_{i=0..n-k} (-1)^i*C(i+k-2, i).

E.g.f.: 4*exp(x) - x - 3. - Elmo R. Oliveira, Aug 08 2024

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A113312 Expansion of (1+x)^2/(1-2x^2+x^3).

Original entry on oeis.org

1, 2, 3, 3, 4, 3, 5, 2, 7, -1, 12, -9, 25, -30, 59, -85, 148, -229, 381, -606, 991, -1593, 2588, -4177, 6769, -10942, 17715, -28653, 46372, -75021, 121397, -196414, 317815, -514225, 832044, -1346265, 2178313, -3524574, 5702891, -9227461, 14930356, -24157813, 39088173, -63245982, 102334159

Offset: 0

Paul Barry, Oct 25 2005

Diagonal sums of A113310.

Index entries for linear recurrences with constant coefficients, signature (0, 2, -1).

LinearRecurrence[{0,2,-1},{1,2,3},50] (* Harvey P. Dale, May 19 2014 *)

a(n)=2a(n-2)-a(n-3); a(n)=sum{k=0..floor(n/2), sum{i=0..n-2k, (-1)^i*C(i+k-2, i)}}.

A113313 Riordan array (1-2x,x/(1-x)).

Original entry on oeis.org

1, -2, 1, 0, -1, 1, 0, -1, 0, 1, 0, -1, -1, 1, 1, 0, -1, -2, 0, 2, 1, 0, -1, -3, -2, 2, 3, 1, 0, -1, -4, -5, 0, 5, 4, 1, 0, -1, -5, -9, -5, 5, 9, 5, 1, 0, -1, -6, -14, -14, 0, 14, 14, 6, 1, 0, -1, -7, -20, -28, -14, 14, 28, 20, 7, 1, 0, -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, 0, -1, -9, -35

Offset: 0

Paul Barry, Oct 25 2005

Row sums are (1,-1,0,0,0,...) = 2*C(0,n) - C(1,n).
Diagonal sums are -2*0^n - F(n-4) with g.f. (1 - 3x + 2x^2) / (1 - x - x^2).
Inverse of A113310.

			The triangle T(n, k) begins:
n\k 0  1  2   3   4   5  6  7  8 9 10 ...
0:  1
1: -2  1
2:  0 -1  1
3:  0 -1  0   1
4:  0 -1 -1   1   1
5:  0 -1 -2   0   2   1
6:  0 -1 -3  -2   2   3  1
7:  0 -1 -4  -5   0   5  4  1
8:  0 -1 -5  -9  -5   5  9  5  1
9:  0 -1 -6 -14 -14   0 14 14  6 1
10: 0 -1 -7 -20 -28 -14 14 28 20 7  1
... Reformatted. - _Wolfdieter Lang_, Jan 06 2015

Cf. A113310.

T(n, k) = C(n-1, n-k) - 2*C(n-2, n-k-1).

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-x + x^3/3!) = -x - 2*x^2/2! - 2*x^3/3! + 5*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014

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A113311 Expansion of (1+x)^2/(1-x).

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A113312 Expansion of (1+x)^2/(1-2x^2+x^3).

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A113313 Riordan array (1-2x,x/(1-x)).

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