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.

A026012 Second differences of Catalan numbers A000108.

Original entry on oeis.org

1, 2, 6, 19, 62, 207, 704, 2431, 8502, 30056, 107236, 385662, 1396652, 5088865, 18642420, 68624295, 253706790, 941630580, 3507232740, 13105289370, 49114150020, 184560753390, 695267483664, 2625197720454, 9933364416572, 37660791173152, 143048202990504
Offset: 0

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Author

N. J. A. Sloane, Clark Kimberling

Keywords

Comments

Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = s(2n) = 2.
Number of Dyck paths of semilength n+2 with no initial and no final UD's. Example: a(2)=6 because the only Dyck paths of semilength 4 with no initial and no final UD's are UUDUDUDD, UUDUUDDD, UUUDDUDD, UUUDUDDD, UUDDUUDD, UUUUDDDD. - Emeric Deutsch, Oct 26 2003
Number of branches of length 1 starting from the root in all ordered trees with n+1 edges. Example: a(1)=2 because the tree /\ has two branches of length 1 starting from the root and the path-tree of length 2 has none. a(n) = Sum_{k=0..n+1} (k*A127158(n+1,k)). - Emeric Deutsch, Mar 01 2007
Number of staircase walks from (0,0) to (n,n) that never cross y=x+2. Example: a(3) = 19 because up,up,up,right,right,right is not allowed but the other binomial(6,3)-1 = 19 paths are. - Mark Spindler, Nov 11 2012
Number of standard Young tableaux of skew shape (n+2,n)/(2), for n>=2. - Ran Pan, Apr 07 2015

References

  • S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 188, 196).

Links

Crossrefs

T(2n, n), where T is the array defined in A026009.
Cf. A000108, A000245, A000344, A039599, A059346, A127158, A342912.

Programs

  • Mathematica
    Differences[Table[CatalanNumber[n], {n, 0, 28}], 2] (* Jean-François Alcover, Sep 28 2012 *)
Table[Binomial[2n,n]-Binomial[2n,n-3],{n,0,26}] (* Mark Spindler, Nov 11 2012 *)
  • PARI
    a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)); /* Joerg Arndt, Aug 19 2012 */

Formula

Expansion of (1+x^1*C^3)*C^1, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)). - Emeric Deutsch, Oct 26 2003
a(n) = Sum_{k=0..2} A039599(n,k) = A000108(n) + A000245(n) + A000344(n). - Philippe Deléham, Nov 12 2008
a(n) = binomial(2*n,n)/(n+1)*hypergeom([-2,n+1/2],[n+2],4). - Peter Luschny, Aug 15 2012
a(n) = binomial(2*n,n) - binomial(2n,n-3). - Mark Spindler, Nov 11 2012
D-finite with recurrence (n+3)*a(n) + (-5*n-6)*a(n-1) + 2*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Jun 20 2013
E.g.f.: exp(2*x)*(BesselI(0,2*x) - BesselI(3,2*x)). - Ilya Gutkovskiy, Feb 28 2017
Sum_{n>=0} a(n)/4^n = 6. - Amiram Eldar, Jul 10 2023
a(n) = C(n+2)+C(n)-2*C(n+1), C = A000108. - Alois P. Heinz, Apr 02 2025
Binomial transform of A342912. - Mélika Tebni, Apr 05 2025

A082397 Number of directed aggregates of height <= 2 with n cells.

Original entry on oeis.org

1, 2, 5, 11, 26, 62, 153, 385, 988, 2573, 6786, 18084, 48621, 131718, 359193, 985185, 2715972, 7521567, 20915256, 58373586, 163462815, 459136809, 1293223230, 3651864606, 10336625731, 29321683082, 83344398533, 237344961291
Offset: 1

Views

Author

Fouad IBN MAJDOUB HASSANI, Apr 14 2003

Keywords

Comments

Conjecture: partial sums of A342912. - Sean A. Irvine, Jul 16 2022

References

  • Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux décalés oscillants. Thèse de Doctorat. 1991. Laboratoire de Recherche en Informatique, Université Paris-Sud XI, France.

Links

Programs

  • Maple
    A082397 := proc(n)
    add( (-1)^(k+1)*binomial(n+1,k+1)*binomial(k,floor((k-1)/2)),k=1..n) ;
end proc:
seq(A082397(n),n=1..30) ; # R. J. Mathar, Jun 27 2022
  • Mathematica
    Table[Sum[(-1)^(i+1)*Binomial[k+1, i+1] Binomial[i, Floor[(i-1)/2]], {i,1,k}], {k,1,20}] (* Rigoberto Florez, Dec 10 2022 *)
  • Python
    import math
def Sum(k):
    S= sum((-1)**(i+1)*math.comb(k,i+1)*math.comb(i,math.floor((i-1)/2)) for i in range(1,k))
    return S
for i in range (2,20): print(Sum(i))
# Rigoberto Florez, Dec 10 2022

Formula

a(n) = Sum_{k=1..n}(-1)^(k+1)*binomial(n+1, k+1)*binomial(k, floor((k-1)/2)). E.g.f.: -exp(x)*int(-BesselI(1, 2*x)+BesselI(2, 2*x), x)-exp(x)*(-BesselI(1, 2*x)+BesselI(2, 2*x)). - Vladeta Jovovic, Sep 18 2003
Conjecture D-finite with recurrence +(n+2)*a(n) +(-3*n-2)*a(n-1) -n*a(n-2) +3*n*a(n-3)=0. - R. J. Mathar, Jun 27 2022

Extensions

More terms from Vladeta Jovovic, Sep 18 2003

A379907 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} (-1)^(n - k - i) * binomial(n - k, i) * binomial(k + 2*i, i) * (k + 1) / (k + 1 + i).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 3, 1, 6, 9, 9, 7, 4, 1, 15, 21, 21, 17, 11, 5, 1, 36, 51, 51, 42, 29, 16, 6, 1, 91, 127, 127, 106, 76, 46, 22, 7, 1, 232, 323, 323, 272, 200, 128, 69, 29, 8, 1, 603, 835, 835, 708, 530, 352, 204, 99, 37, 9, 1, 1585, 2188, 2188, 1865, 1415, 965, 587, 311, 137, 46, 10, 1
Offset: 0

Views

Author

Werner Schulte, Jan 05 2025

Keywords

Comments

Conjecture: Let A = (g(t), f(t)) and B = (u(t), v(t)) be (triangular) Riordan arrays with A(n, k) = [t^n](g(t)*(f(t))^k) and B(n, k) = [t^n](u(t)*(v(t))^k). Then T = (g(t)*u(f(t)), v(f(t))*t/f(t)) is the Riordan array with T(n, k) = [t^n](g(t)*u(f(t))*(v(f(t))*t/f(t))^k) = Sum_{i=0..n-k} A(n-k, i) * B(k+i, k) for 0 <= k <= n.

Examples

			Triangle T(n, k) for 0 <= k <= n starts:
n \k :     0     1     2     3     4    5    6    7    8   9  10  11
====================================================================
   0 :     1
   1 :     0     1
   2 :     1     1     1
   3 :     1     2     2     1
   4 :     3     4     4     3     1
   5 :     6     9     9     7     4    1
   6 :    15    21    21    17    11    5    1
   7 :    36    51    51    42    29   16    6    1
   8 :    91   127   127   106    76   46   22    7    1
   9 :   232   323   323   272   200  128   69   29    8   1
  10 :   603   835   835   708   530  352  204   99   37   9   1
  11 :  1585  2188  2188  1865  1415  965  587  311  137  46  10   1
  etc.

Crossrefs

Cf. A005043 (column 0), A001006 (column 1 and 2), A102071 (column 3).
Cf. A000108, A342912 (row sums), A379824 (alternating row sums), A379823 (central terms).

Programs

  • Maple
    gf := 2/(sqrt((1-3*t)*(t+1)) - 2*(t+1)*t*x + t+1): ser := simplify(series(gf,t,12)):
ct := n -> coeff(ser,t,n): row := n -> local k; seq(coeff(ct(n), x, k), k = 0..n):
seq(row(n), n = 0..11);  # Peter Luschny, Jan 05 2025
  • PARI
    T(n,k) = sum(i=0,n-k,(-1)^(n-k-i)*binomial(n-k,i)*binomial(k+2*i,i)*(k+1)/(k+1+i))
  • PARI
    T(n,k)=polcoef(polcoef(2/(sqrt((1-3*t)*(1+t))+(1+t)*(1-2*x*t))+x*O(x^k),k,x)+t*O(t^n),n,t);
       m=matrix(15,15,n,k,if(k>n,0,T(n-1,k-1)))

Formula

Riordan array (C(t/(1+t)) / (1+t), t * C(t/(1+t))) where C(x) is g.f. of A000108.
Riordan array ((1 + t - sqrt(1 - 2*t - 3*t^2))/(2*t*(1 + t)), (1 + t - sqrt(1-2*t-3*t^2))/2).
G.f.: 2/(sqrt((1 - 3*t)*(t + 1)) - 2*(t + 1)*t*x + t + 1).
Conjecture: T(n, k) = T(n, k-1) + T(n-1, k-1) - T(n-1, k-2) - T(n-2, k-2) for 2 <= k <= n.
T(n, k) = (-1)^(k-n)*hypergeom([k-n, k/2+1, (k+1)/2], [1, k + 2], 4). - Peter Luschny, Jan 06 2025
Showing 1-3 of 3 results.

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

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