A124788 -id:A124788 - cp's OEIS frontend

A124789 Expansion of (1+x^2)/(1-x^4-x^5).

1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 17, 21, 27, 31, 33, 38, 48, 58, 64, 71, 86, 106, 122, 135, 157, 192, 228, 257, 292, 349, 420, 485, 549, 641, 769, 905, 1034, 1190, 1410, 1674, 1939, 2224, 2600, 3084, 3613

Offset: 0

Paul Barry, Nov 07 2006

Diagonal sums of A124788.

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

Cf. A017827, A103372.

CoefficientList[Series[(1+x^2)/(1-x^4-x^5),{x,0,60}],x] (* or *) LinearRecurrence[ {0,0,0,1,1},{1,0,1,0,1},60] (* Harvey P. Dale, Aug 20 2013 *)

a(n) = Sum_{k=0..floor(n/2)} C(floor(k/2),n-2*k).
a(n) = A017827(n)+A017827(n-2). - R. J. Mathar, May 09 2013
a(n) = A103372(n-3) for n >= 4. - Georg Fischer, Nov 03 2018
a(n) = (-1)^n*A124746(n). - R. J. Mathar, Jun 30 2020

A124790 A generalized Motzkin triangle.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 3, 4, 3, 2, 1, 0, 6, 9, 6, 5, 2, 1, 0, 15, 21, 15, 12, 6, 3, 1, 0, 36, 51, 36, 30, 15, 9, 3, 1, 0, 91, 127, 91, 76, 40, 25, 10, 4, 1, 0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1

Offset: 0

Paul Barry, Nov 07 2006

Columns include A005043, A001006, A002026. Row sums are A124791. For even k, column k has g.f. x^k*M(x)^(k/2), where M(x)=2/(1-x+sqrt(1-2x-3x^2)) is the g.f. of A001006. For odd k, column k has g.f. x^k*S(x)*M(x)^floor(k/2), S(x)=(1+x-sqrt(1-2x-3x^2))/(2x(1+x)), the g.f. of A005043.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 1, 2, 1, 1,
0, 3, 4, 3, 2, 1,
0, 6, 9, 6, 5, 2, 1,
0, 15, 21, 15, 12, 6, 3, 1,
0, 36, 51, 36, 30, 15, 9, 3, 1,
0, 91, 127, 91, 76, 40, 25, 10, 4, 1,
0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1
Production matrix begins
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1
- _Paul Barry_, Apr 07 2011

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Triangle is the product of A124788 and A124305, that is, it is the product of the expansion of (1+x*y)/(1-x^2*y^2-x^3*y^2) and the inverse of the Riordan array (1,x(1-x^2)).

A124816 Product of Riordan array (1,x(1-x^2))^(-1) and number triangle T(n,k)=C(floor(k/2),n-k).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 0, 3, 3, 3, 2, 1, 0, 0, 7, 4, 5, 2, 1, 0, 12, 12, 12, 10, 6, 3, 1, 0, 0, 30, 18, 24, 12, 9, 3, 1, 0, 55, 55, 55, 50, 32, 22, 10, 4, 1, 0, 0, 143, 88, 121, 66, 57, 25, 14, 4, 1, 0, 273, 273, 273

Offset: 0

Paul Barry, Nov 08 2006

Product of A124305 and A124788. Columns include aerated A001764,A047749(n+1),A124817,A084081. Row sums are A124818.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 0, 2, 1, 1,
0, 3, 3, 3, 2, 1,
0, 0, 7, 4, 5, 2, 1,
0, 12, 12, 12, 10, 6, 3, 1,
0, 0, 30, 18, 24, 12, 9, 3, 1,
0, 55, 55, 55, 50, 32, 22, 10, 4, 1

Cf. A124790.

A127829 Number triangle mod(C(floor(k/2),n-k),2).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1

Offset: 0

Paul Barry, Feb 01 2007

Row sums are A127830. Inverse is A127831.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 0, 1, 1,
0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 0, 0, 1, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1

T(n,k)=mod(A124788(n,k),2)

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A124789 Expansion of (1+x^2)/(1-x^4-x^5).

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A124790 A generalized Motzkin triangle.

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A124816 Product of Riordan array (1,x(1-x^2))^(-1) and number triangle T(n,k)=C(floor(k/2),n-k).

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A127829 Number triangle mod(C(floor(k/2),n-k),2).

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