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.
1 -1, 1 1, -2, 2 -1, 3, -6, 6 1, -4, 12, -24, 6 -1, 5, -20, 60, -30, 30 1, -6, 30, -120, 90, -180, 20 -1, 7, -42, 210, -210, 630, -140, 140 1, -8, 56, -336, 420, -1680, 560, -1120, 70
a := proc(n,k) (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)! end: seq(print(seq(a(n,k),k=0..n)),n=0..8);
t[n_, k_] := (-1)^(n - k)*Floor[k/2]!^(-2)*n!/(n - k)!; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)
for(n=0,10, for(k=0,n, print1((-1)^(n -k)*( (floor(k/2))! )^(-2)*(n!/(n - k)!), ", "))) \\ G. C. Greubel, Aug 01 2017
LinearRecurrence[{0,1,0,2},{1,0,-1,1},70] (* Harvey P. Dale, Sep 04 2024 *)
Triangle begins: 1; -1, 6; 1, -12, 30; -1, 18, -90, 140; 1, -24, 180, -560, 630; -1, 30, -300, 1400, -3150, 2772; 1, -36, 450, -2800, 9450, -16632, 12012;
swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end: a := proc(n, k) (-1)^(n-k)*binomial(n,k)*swing(2*k+1) end: seq(print(seq(a(n,k),k=0..n)),n=0..8);
T[n_,k_] := ((-1)^(Mod[k,2]+n)*((2*k+1)!/(k!)^2)*Binomial[n,n-k]);
Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* Detlef Meya, Oct 07 2023 *)
a000007 = (0 ^) a000007_list = 1 : repeat 0 -- Reinhard Zumkeller, May 07 2012, Mar 27 2012
[1] cat [0:n in [1..100]]; // Sergei Haller, Dec 21 2006
A000007 := proc(n) if n = 0 then 1 else 0 fi end: seq(A000007(n), n=0..20); spec := [A, {A=Z} ]: seq(combstruct[count](spec, size=n+1), n=0..20);
Table[If[n == 0, 1, 0], {n, 0, 99}]
Table[Boole[n == 0], {n, 0, 99}] (* Michael Somos, Aug 25 2012 *)
Join[{1},LinearRecurrence[{1},{0},102]] (* Ray Chandler, Jul 30 2015 *)
PadRight[{1},120,0] (* Harvey P. Dale, Jul 18 2024 *)
{a(n) = !n};
def A000007(n): return int(n==0) # Chai Wah Wu, Feb 04 2022
G.f. = x + 5*x^2 + 14*x^3 + 30*x^4 + 55*x^5 + 91*x^6 + 140*x^7 + 204*x^8 + ...
List([0..30], n-> n*(n+1)*(2*n+1)/6); # G. C. Greubel, Dec 31 2019
a000330 n = n * (n + 1) * (2 * n + 1) `div` 6 a000330_list = scanl1 (+) a000290_list -- Reinhard Zumkeller, Nov 11 2012, Feb 03 2012
[n*(n+1)*(2*n+1)/6: n in [0..50]]; // Wesley Ivan Hurt, Jun 28 2014
[0] cat [((2*n+3)*Binomial(n+2,2))/3: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014
A000330 := n -> n*(n+1)*(2*n+1)/6; a := n->(1/6)*n*(n+1)*(2*n+1): seq(a(n),n=0..53); # Emeric Deutsch with(combstruct): ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m*2), m=1..45) ; # Zerinvary Lajos, Jan 02 2008 nmax := 44; for n from 0 to nmax do fz(n) := product( (1-(2*m-1)*z)^(n+1-m) , m=1..n); c(n) := abs(coeff(fz(n),z,1)); end do: a := n-> c(n): seq(a(n), n=0..nmax); # Johannes W. Meijer, Mar 07 2009
Table[Binomial[w+2, 3] + Binomial[w+1, 3], {w, 0, 30}]
CoefficientList[Series[x(1+x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)
Accumulate[Range[0,50]^2] (* Harvey P. Dale, Sep 25 2014 *)
A000330(n):=binomial(n+2,3)+binomial(n+1,3)$ makelist(A000330(n),n,0,20); /* Martin Ettl, Nov 12 2012 */
{a(n) = n * (n+1) * (2*n+1) / 6};
upto(n) = [x*(x+1)*(2*x+1)/6 | x<-[0..n]] \\ Cino Hilliard, Jun 18 2007, edited by M. F. Hasler, Jan 02 2024
a=lambda n: (n*(n+1)*(2*n+1))//6 # Indranil Ghosh, Jan 04 2017
[n*(n+1)*(2*n+1)/6 for n in (0..30)] # G. C. Greubel, Dec 31 2019
First five rows: 1 1...1 1...3...1 1...5...4...1 1...7...9...5...1 First five polynomials u(n,x): 1 1 + x 1 + 3x + x^2 1 + 5x + 4x^2 + x^3 1 + 7x + 9x^2 + 5x^3 + x^4
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A029653 *)
from sympy import Poly from sympy.abc import x def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1 def a(n): return Poly(u(n, x), x).all_coeffs()[::-1] for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017
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