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 1-6 of 6 results.

A060921 Bisection of Fibonacci triangle A037027: odd-indexed members of column sequences of A037027 (not counting leading zeros).

Original entry on oeis.org

1, 3, 2, 8, 10, 3, 21, 38, 22, 4, 55, 130, 111, 40, 5, 144, 420, 474, 256, 65, 6, 377, 1308, 1836, 1324, 511, 98, 7, 987, 3970, 6666, 6020, 3130, 924, 140, 8, 2584, 11822, 23109, 25088, 16435, 6588, 1554, 192, 9
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Row sums give A002450. Column sequences (without leading zeros) give for m=0..5: A001906, 2*A001870, A061182, 4*A061183, A061184, 2*A061185.
Companion triangle (odd-indexed members) A060920.

Examples

			{1}; {3,2}; {8,10,3}; {21,38,22,4}; ...; pFo(2,x) = 2*(1-x).

Links

Formula

a(n, m) = A037027(2*n+1-m, m).
a(n, m) = (2*(n-m+1)*A060920(n, m-1)+2*(2*n+1)*a(n-1, m-1))/(5*m), n >= m>0; a(n, 0) := S(n, 3)=A001906(n+1) with Chebyshev's S(n, x) polynomials A049310; else 0.
G.f. for column m >= 0: x^m*pFo(m+1, x)/(1-3*x+x^2)^(m+1), where pFo(n, x) := Sum_{m=0..n-1} A061177(n-1, m)*x^m (row polynomials of signed triangle A061177).
G.f.: 1/(1 - (3+2*y)*x + (1+y)^2*x^2). - Vladeta Jovovic, Oct 11 2003

A061176 Coefficients of polynomials ( (1 -x +sqrt(x))^n + (1 -x -sqrt(x))^n )/2.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, 0, 0, -1, 1, 2, -5, 2, 1, 1, 5, -15, 15, -5, -1, 1, 9, -30, 41, -30, 9, 1, 1, 14, -49, 77, -77, 49, -14, -1, 1, 20, -70, 112, -125, 112, -70, 20, 1, 1, 27, -90, 126, -117, 117, -126, 90, -27, -1, 1, 35, -105, 90, 45, -131, 45, 90, -105, 35, 1
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

The row polynomial pFe(k+1, x) = Sum_{j=0..k+1} T(k+1, j)*x^j is the numerator of the g.f. for the k-th column sequence of A060920, the even part of the bisected Fibonacci triangle.

Examples

			The first few polynomials are:
pFe(0,x) = 1.
pFe(1,x) = 1 -   x.
pFe(2,x) = 1 -   x +   x^2.
pFe(3,x) = 1 - 0*x + 0*x^2 -   x^3.
pFe(4,x) = 1 + 2*x - 5*x^2 + 2*x^3 + x^4.
Number triangle begins as:
  1;
  1, -1;
  1, -1,   1;
  1,  0,   0,  -1;
  1,  2,  -5,   2,    1;
  1,  5, -15,  15,   -5,  -1;
  1,  9, -30,  41,  -30,   9,   1;
  1, 14, -49,  77,  -77,  49, -14, -1;
  1, 20, -70, 112, -125, 112, -70, 20, 1;

Links

Crossrefs

Cf. A059841, A060920, A061177 (companion triangle), A180957.

Programs

  • Magma
    A061176:= func< n,k | (&+[(-1)^(k+j)*Binomial(n,2*j)*Binomial(n-2*j,k-j): j in [0..k]]) >;
[A061176(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 06 2021
  • Mathematica
    T[n_, k_]:= Sum[(-1)^(k+j)*Binomial[n, 2*j]*Binomial[n-2*j, k-j], {j,0,k}];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)
  • Sage
    def A061176(n,k): return sum((-1)^(k+j)*binomial(n,2*j)*binomial(n-2*j,k-j) for j in (0..k))
flatten([[A061176(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 06 2021

Formula

T(n, k) = coefficients of x^k of ((1-x+sqrt(x))^n + (1-x-sqrt(x))^n)/2.
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n, 2*j)*binomial(n-2*j, k-j), if 0 <= k <= floor(n/2) and T(n, k) = (-1)^n*T(n, n-k) if floor(n/2) < k <= n, otherwise 0.
Sum_{k=0..n} T(n, k) = A059841(n) = (1 + (-1)^n)/2. - G. C. Greubel, Apr 06 2021

A061182 Third column (m=2) of triangle A060921 (bisection of Fibonacci triangle, odd part).

Original entry on oeis.org

3, 22, 111, 474, 1836, 6666, 23109, 77378, 252177, 804228, 2519640, 7777860, 23709783, 71501422, 213619683, 633011454, 1862264196, 5443487406, 15820188729, 45739697306, 131624104677, 377157259848
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator polynomial is sum(A061177(2,m)*x^m,m=0..2).

Crossrefs

A001906, 2*A001870.

Formula

a(n)= A060921(n+2, 2).
G.f.: (3*(1+x^2)-5*x)/(1-3*x+x^2)^3.

A061183 One-fourth of the fourth (m=3) column of triangle A060921 (bisection of Fibonacci triangle, odd part).

Original entry on oeis.org

1, 10, 64, 331, 1505, 6272, 24540, 91527, 328768, 1145650, 3893630, 12958400, 42364427, 136389128, 433263360, 1360269093, 4226523495, 13011186624, 39722775806, 120366164765, 362255552384, 1083513943700
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator polynomial of g.f. is (1/4) * Sum_{m=0..3} A061177(3,m)*x^m.

Links

Crossrefs

Cf. A060921, A061177, A061182.

Programs

  • Mathematica
    CoefficientList[Series[((1-x^3)-2(x-x^2))/(1-3x+x^2)^4,{x,0,30}],x] (* or *) LinearRecurrence[{12,-58,144,-195,144,-58,12,-1},{1,10,64,331,1505,6272,24540,91527},30] (* Harvey P. Dale, Jun 17 2022 *)

Formula

a(n) = A060921(n+3, 3)/4.
G.f.: ((1-x^3)-2*(x-x^2))/(1-3*x+x^2)^4.

A061184 Fifth (m=4) column of triangle A060921 (bisection of Fibonacci triangle, odd part).

Original entry on oeis.org

5, 65, 511, 3130, 16435, 77645, 339535, 1399478, 5504650, 20845300, 76495450, 273381350, 955187033, 3272875935, 11024814945, 36584603310, 119796766005, 387639512331, 1240994295715, 3934750789180
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator polynomial of g.f. is sum(A061177(4,m)*x^m,m=0..4).

Crossrefs

4*A061183.

Formula

a(n)= A060921(n+4, 4).
G.f.: (5*(1+x^4)-10*(x+x^3)+11*x^2)/(1-3*x+x^2)^5.

A061185 One half of sixth (m=5) column of triangle A060921 (bisection of Fibonacci triangle, odd part).

Original entry on oeis.org

3, 49, 462, 3294, 19715, 104517, 506646, 2292310, 9817920, 40210800, 158677370, 606790410, 2258770689, 8214432303, 29269938510, 102434633406, 352793077413, 1197764971911, 4014411070092
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator polynomial of g.f. is sum(A061177(5,m)*x^m,m=0..5)/2.

Crossrefs

A061184.

Formula

a(n)= A060921(n+5, 5)/2.
G.f.: (3*(1-x^5)-5*(x-x^4)+3*(x^2-x^3))/(1-3*x+x^2)^6.
Showing 1-6 of 6 results.

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