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 1-3 of 3 results.

A182015 Diagonal sums of triangle A182013.

Original entry on oeis.org

1, 2, 5, 11, 26, 60, 145, 353, 884, 2241, 5786, 15108, 39941, 106558, 286809, 777505, 2121668, 5822287, 16059288, 44494738, 123782207, 345615047, 968211110, 2720561790, 7665640267, 21654105734, 61312389677, 173978404587, 494667697706, 1409099662020
Offset: 0

Views

Author

Emanuele Munarini, Apr 06 2012

Keywords

Crossrefs

Cf. A182013.

Programs

  • Mathematica
    M[n_]:=If[n==0,1,Coefficient[(1+x+x^2)^(n+1),x^n]/(n+1)]; Table[Sum[(n-i+1)M[i],{i,0,n}]-Sum[(n-2i)M[i],{i,0,Floor[n/2]}],{n,0,30}]
  • Maxima
    M(n):=coeff(expand((1+x+x^2)^(n+1)),x^n)/(n+1);
makelist(sum((n-i+1)*M(i),i,0,n)-sum((n-2*i)*M(i),i,0,floor(n/2)),n,0,30);

Formula

a(n) = sum(sum(M(i),i=k..n-k),k=0..n), where the M(n)'s are the Motzkin numbers.
a(n) = sum((n-i+1)*M(i),i=0..n) - sum((n-2*i)*M(i),i=0..floor(n/2)).
G.f.: (1-x+x*sqrt(1-2*x-3*x^2)-sqrt(1-2*x^2-3*x^4))/(2*x^3*(1-x)^2).

A182016 Central coefficients of triangle A182013.

Original entry on oeis.org

1, 3, 15, 85, 531, 3545, 24833, 180251, 1344163, 10237001, 79287469, 622596063, 4945078477, 39658283943, 320689293119, 2611817435525, 21405148148051, 176396318594937, 1460795339822349, 12150467810552223, 101463416752305385, 850306144532133787
Offset: 0

Views

Author

Emanuele Munarini, Apr 06 2012

Keywords

Programs

  • Mathematica
    M[n_]:=If[n==0,1,Coefficient[(1+x+x^2)^(n+1),x^n]/(n+1)]; Table[Sum[M[i],{i,n,2n}],{n,0,40}]
  • Maxima
    M(n):=coeff(expand((1+x+x^2)^(n+1)),x^n)/(n+1);
makelist(sum(M(i),i,n,2*n),n,0,20);

Formula

a(n) = sum(M(i),i=n..2*n), where the M(n)'s are the Motzkin numbers.
Recurrence: a(n+1) = a(n) + M(2n+2) + M(2n+1) - M(n).

A182017 Row square-sums of triangle A182013.

Original entry on oeis.org

1, 5, 29, 165, 1020, 6606, 44805, 314299, 2266834, 16714004, 125501364, 956725836, 7387322749, 57669478609, 454492588153, 3611698593169, 28911762536992, 232949503809562, 1887883708627582, 15380196764853214, 125893997550676627, 1034945699861044243
Offset: 0

Views

Author

Emanuele Munarini, Apr 06 2012

Keywords

Crossrefs

Cf. A001006, A182013.

Programs

  • Mathematica
    M[n_]:=If[n==0,1,Coefficient[(1+x+x^2)^(n+1),x^n]/(n+1)]; Table[Sum[Sum[M[i],{i,k,n}]^2,{k,0,n}],{n,0,40}]
  • Maxima
    M(n):=coeff(expand((1+x+x^2)^(n+1)),x^n)/(n+1);
makelist(sum(sum(M(i),i,k,n)^2,k,0,n),n,0,20);

Formula

a(n) = Sum_{k=0..n} (Sum_{i=k..n} M(i))^2, where the M(n)'s are the Motzkin numbers (A001006).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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