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.

A167772 Riordan array (c(x)/(1+x*c(x)), x*c(x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 6, 8, 6, 3, 1, 18, 24, 18, 10, 4, 1, 57, 75, 57, 33, 15, 5, 1, 186, 243, 186, 111, 54, 21, 6, 1, 622, 808, 622, 379, 193, 82, 28, 7, 1, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1, 7338, 9458, 7338, 4596, 2476, 1164, 474, 163, 45, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Nov 11 2009, corrected Nov 12 2009

Keywords

Examples

			Triangle begins:
     1;
     0,    1;
     1,    1,    1;
     2,    3,    2,    1;
     6,    8,    6,    3,   1;
    18,   24,   18,   10,   4,   1;
    57,   75,   57,   33,  15,   5,   1;
   186,  243,  186,  111,  54,  21,   6,  1;
   622,  808,  622,  379, 193,  82,  28,  7,  1;
  2120, 2742, 2120, 1312, 690, 311, 118, 36,  8,  1;
Production matrix begins:
  0, 1;
  1, 1, 1;
  1, 1, 1, 1;
  1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ... - _Philippe Deléham_, Mar 03 2013

Links

Crossrefs

Cf. A000957, A000958 (row sums), A001558, A001559, A104629, A167769.
Diagonals: A000012, A000217, A001477, A166830.

Programs

  • Haskell
    import Data.List (genericIndex)
a167772 n k = genericIndex (a167772_row n) k
a167772_row n = genericIndex a167772_tabl n
a167772_tabl = [1] : [0, 1] :
               map (\xs@(:x:) -> x : xs) (tail a065602_tabl)
-- Reinhard Zumkeller, May 15 2014
  • Mathematica
    A065602[n_, k_]:= A065602[n,k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n-j) -k-1), {j, 0, (n-k)/2}];
T[n_, k_]:= If[k==0, A065602[n+1,3] + Boole[n==0], A065602[n+1, k+1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2022 *)
  • SageMath
    def A065602(n,k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
def A167772(n,k):
    if (k==0): return A065602(n+1,3) + bool(n==0)
    else: return A065602(n+1,k+1)
flatten([[A167772(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2022

Formula

Sum_{k=0..n} T(n, k) = A000958(n+1).
From Philippe Deléham, Nov 12 2009: (Start)
Sum_{k=0..n} T(n,k)*2^k = A014300(n).
Sum_{k=0..n} T(n,k)*2^(n-k) = A064306(n). (End)
For n > 0: T(n,0) = A065602(n+1,3), T(n,k) = A065602(n+1,k+1), k = 1..n. - Reinhard Zumkeller, May 15 2014

A167769 Pendular trinomial triangle (p=0), read by rows of 2n+1 terms (n>=0), defined by the recurrence : if 0

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 6, 8, 6, 3, 1, 0, 0, 1, 4, 10, 18, 24, 18, 10, 4, 1, 0, 0, 1, 5, 15, 33, 57, 75, 57, 33, 15, 5, 1, 0, 0, 1, 6, 21, 54, 111, 186, 243, 186, 111, 54, 21, 6, 1, 0, 0, 1, 7, 28, 82, 193, 379, 622, 808, 622, 379, 193, 82, 28, 7, 1, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Nov 11 2009

Keywords

Comments

See A119369 for p=1 and A122445 for p=2. The diagonals may be generated by iterated convolutions of a base sequence B (A000108(n)) with the sequence C (A000957(n+1)) of central terms.

Examples

			Triangle begins :
  1;
  1, 0,  0;
  1, 1,  1,  0,  0;
  1, 2,  3,  2,  1,  0,  0;
  1, 3,  6,  8,  6,  3,  1,  0,  0;
  1, 4, 10, 18, 24, 18, 10,  4,  1, 0, 0,
  1, 5, 15, 33, 57, 75, 57, 33, 15, 5, 1, 0, 0; ...

References

  • Kim, Ki Hang; Rogers, Douglas G.; Roush, Fred W. Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577--594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013) - From N. J. A. Sloane, Jun 05 2012

Links

Crossrefs

Cf. A000108, A000957, A000958, A001558, A001559, A071724, A104629.

Programs

  • Maple
    T:= proc(n, k) option remember;
      if k=0 and n=0 then 1;
    elif k<0 or k>2*(n-1) then 0;
    elif n=2 and k<3 then 1;
    elif kG. C. Greubel, Mar 17 2021
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 && n==0, 1, If[k<0 || k>2*(n-1), 0, If[n==2 && k<3, 1, If[kG. C. Greubel, Mar 17 2021 *)
  • PARI
    T(n, k)=if(k==0 && n==0, 1, if(k>2*n-2 || k<0, 0, if(n==2 && k<=2, 1, if(kPaul D. Hanna, Nov 12 2009
  • Sage
    @CachedFunction
def T(n, k):
    if (k==0 and n==0): return 1
    elif (k<0 or k>2*(n-1)): return 0
    elif (n==2 and k<3): return 1
    elif (kG. C. Greubel, Mar 17 2021

Formula

Sum_{k=0..2*n} T(n,k) = A071724(n) = [n=0] + 3*binomial(2n,n-1)/(n+2) = [n=0] + n*C(n)/(n+2), where C(n) are the Catalan numbers (A000108). - G. C. Greubel, Mar 17 2021

A237619 Riordan array (1/(1+x*c(x)), x*c(x)) where c(x) is the g.f. of Catalan numbers (A000108).

Original entry on oeis.org

1, -1, 1, 0, 0, 1, -1, 1, 1, 1, -2, 2, 3, 2, 1, -6, 6, 8, 6, 3, 1, -18, 18, 24, 18, 10, 4, 1, -57, 57, 75, 57, 33, 15, 5, 1, -186, 186, 243, 186, 111, 54, 21, 6, 1, -622, 622, 808, 622, 379, 193, 82, 28, 7, 1, -2120, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 10 2014

Keywords

Examples

			Triangle begins:
    1;
   -1,  1;
    0,  0,  1;
   -1,  1,  1,  1;
   -2,  2,  3,  2,  1;
   -6,  6,  8,  6,  3,  1;
  -18, 18, 24, 18, 10,  4, 1;
  -57, 57, 75, 57, 33, 15, 5, 1;
Production matrix begins:
  -1, 1;
  -1, 1, 1;
  -1, 1, 1, 1;
  -1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1, 1;

Links

Crossrefs

Diagonals: A000012, A000217, A023443, A166830.
Cf. A000957, A000958, A001558, A001559, A104629, A126983.
Cf. A065602, A167772.

Programs

  • Mathematica
    A065602[n_, k_]:= A065602[n, k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n - j) -k-1), {j,0,(n-k)/2}];
T[n_, k_]:= If[k==0, A065602[n, 0], If[n==1 && k==1, 1, A065602[n, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 27 2022 *)
  • SageMath
    def A065602(n, k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
def A237619(n, k):
    if (n<2): return (-1)^(n-k)
    elif (k==0): return A065602(n, 0)
    else: return A065602(n, k)
flatten([[A237619(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2022

Formula

Sum_{k=0..n} T(n,k)*x^k = A126983(n), A000957(n+1), A026641(n) for x = 0, 1, 2 respectively.
T(n, k) = A167772(n-1, k-1) for k > 0, with T(n, 0) = A167772(n, 0).
T(n, 0) = A126983(n).
T(n+1, 1) = A000957(n+1).
T(n+2, 2) = A000958(n+1).
T(n+3, 3) = A104629(n) = A000957(n+3).
T(n+4, 4) = A001558(n).
T(n+5, 5) = A001559(n).
T(n, k) = A065602(n, k) for k > 0, with T(n, k) = (-1)^(n-k), for n < 2, and T(n, 0) = A065602(n, 0). - G. C. Greubel, May 27 2022
Showing 1-3 of 3 results.

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

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