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.

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A202551 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, -1, 0, -1, 1, -1, 1, 1, -1, -1, 3, -2, -1, 1, 0, 2, -5, 3, 1, -1, 1, -2, -2, 7, -4, -1, 1, 1, -5, 7, 1, -9, 5, 1, -1, 0, -3, 12, -15, 1, 11, -6, -1, 1, -1, 3, 3, -21, 26, -4, -13, 7, 1, -1
Offset: 0

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Author

Philippe Deléham, Dec 21 2011

Keywords

Comments

Riordan array (1/(1-x+x^2), x*(x-1)/(1-x+x^2)).

Examples

			Triangle begins :
1
1, -1
0, -1, 1
-1, 1, 1, -1
-1, 3, -2, -1, 1
0, 2, -5, 3, 1, -1

Crossrefs

Cf. A129267, A199324

Formula

T(n,k) = T(n-1,k) + T(n-2,k-1) - T(n-1,k-1) - T(n-2,k).
G.f.: 1/(1+(y-1)*x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190873(n+1), A190871(n+1), A057086(n), A057085(n+1), A057084(n), A030240(n), A030192(n), A030191(n), A001787(n+1), A057083(n), A099087(n), A010892(n), A000007(n), (-1)^n*A000045(n+1) for x = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2 respectively.

A360766 a(0) = 0; a(n) = ( (n + sqrt(n))^n - (n - sqrt(n))^n )/(2 * sqrt(n)).

Original entry on oeis.org

0, 1, 4, 30, 320, 4400, 73872, 1462552, 33325056, 858283776, 24641000000, 779935205984, 26972930949120, 1011642325897216, 40890444454377728, 1771640957790000000, 81896889467638120448, 4022826671022707900416, 209224123984489179202560
Offset: 0

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Author

Seiichi Manyama, Mar 11 2023

Keywords

Crossrefs

Main diagonal of A361290.
Cf. A007070, A030192, A093145, A154237.
Cf. A084062.

Programs

  • PARI
    a(n) = polcoeff(lift(Mod('x, ('x-n)^2-n)^n), 1); \\ Kevin Ryde, Mar 16 2023

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} n^(n-1-k) * binomial(n,2*k+1).
a(n) = [x^n] x/(1 - 2*n*x + (n-1)*n*x^2).
a(n) = n! * [x^n] exp(n * x) * sinh(sqrt(n) * x) / sqrt(n) for n > 0.

A276265 Expansion of (1 + 2*x)/(1 - 6*x + 6*x^2).

Original entry on oeis.org

1, 8, 42, 204, 972, 4608, 21816, 103248, 488592, 2312064, 10940832, 51772608, 244990656, 1159308288, 5485905792, 25959585024, 122842075392, 581294942208, 2750717200896, 13016533552128, 61594898107392, 291470187331584, 1379251735345152, 6526689288081408, 30884625316417536
Offset: 0

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Author

Ilya Gutkovskiy, Aug 26 2016

Keywords

Comments

Satisfies recurrence relations system a(n) = 4*a(n-1) + 2*b(n-1), b(n) = 2*b(n-1) + a(n-1), a(0)=1, b(0)=2.
More generally, for the recurrence relations system a(n) = 4*a(n-1) + 2*b(n-1), b(n) = 2*b(n-1) + a(n-1), a(0)=k, b(0)=m solution is a(n) = (((sqrt(3) - 1)*k - 2*m)*(3 - sqrt(3))^n + (sqrt(3)*k + k + 2*m)*(3 + sqrt(3))^n)/(2*sqrt(3)), b(n) = ((-k + sqrt(3)*m + m)*(3 - sqrt(3))^n + (k + (sqrt(3) - 1)*m)*(3 + sqrt(3))^n)/(2*sqrt(3)).
Convolution of A030192 and {1, 2, 0, 0, 0, 0, 0, ...}.

Links

Crossrefs

Cf. A030192, A165663.

Programs

  • Maple
    a:=series((1+2*x)/(1-6*x+6*x^2),x=0,25): seq(coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    LinearRecurrence[{6, -6}, {1, 8}, 25]
CoefficientList[Series[(1 + 2 x)/(1 - 6 x + 6 x^2), {x, 0, 24}], x] (* Michael De Vlieger, Aug 26 2016 *)
  • PARI
    Vec((1+2*x)/(1-6*x+6*x^2) + O(x^99)) \\ Altug Alkan, Aug 26 2016

Formula

O.g.f.: (1 + 2*x)/(1 - 6*x + 6*x^2).
E.g.f.: (5*sqrt(3)*sinh(sqrt(3)*x) + 3*cosh(sqrt(3)*x))*exp(3*x)/3.
a(n) = 6*a(n-1) - 6*a(n-2).
a(n) = ((-5 + sqrt(3))*(3 - sqrt(3))^n + (5 + sqrt(3))*(3 + sqrt(3))^n)/(2*sqrt(3)).
Lim_{n->infinity} a(n+1)/a(n) = 3 + sqrt(3) = A165663.
a(n) = A030192(n)+2*A030192(n-1). - R. J. Mathar, Jan 25 2023
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