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.

Previous Showing 11-18 of 18 results.

A027290 a(n) = Sum_{k=0..floor(n/2)-2} T(n,k) * T(n,k+3), with T given by A026009.

Original entry on oeis.org

3, 9, 73, 230, 1335, 4376, 22144, 74766, 351558, 1213965, 5458347, 19181130, 83735127, 298391598, 1276135088, 4599546922, 19380537318, 70514368558, 293838981413, 1077617352348, 4452600188602, 16440039884624, 67481063581593, 250610375841434, 1023303550651792
Offset: 4

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027291 a(n) = Sum_{k=0..floor(n/2)+1} (k+1) * A026009(n, k).

Original entry on oeis.org

1, 3, 8, 16, 39, 77, 179, 354, 799, 1584, 3504, 6960, 15174, 30183, 65085, 129598, 277071, 552140, 1172348, 2337624, 4935586, 9845978, 20691318, 41292196, 86432614, 172538352, 359934704, 718680768, 1494861996, 2985378027
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027292 a(n) = Sum_{k=0..m} (k+1) * A026009(n, m-k) where m = floor(n/2)+1.

Original entry on oeis.org

1, 3, 8, 12, 31, 48, 121, 192, 475, 768, 1872, 3072, 7398, 12288, 29295, 49152, 116179, 196608, 461284, 786432, 1833202, 3145728, 7290818, 12582912, 29014046, 50331648, 115521296, 201326592, 460153644, 805306368, 1833594171
Offset: 0

Views

Author

Clark Kimberling

Keywords

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

Views

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

A026016 a(n) = binomial(2*n-1, n) - binomial(2*n-1, n+3).

Original entry on oeis.org

1, 3, 10, 34, 117, 407, 1430, 5070, 18122, 65246, 236436, 861764, 3157325, 11622015, 42961470, 159419670, 593636670, 2217608250, 8308432140, 31212003420, 117544456770, 443690433654, 1678353186780, 6361322162444, 24155384502452, 91882005146652
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Number of (s(0), s(1), ..., s(2n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n-1) = 3. Also a(n) = T(2n-1,n-1), where T is the array defined in A026009.
Number of integer lattice paths from (0,2) to (n-1,n+2) that do not cross the main diagonal.

Links

Crossrefs

Cf. A000108, A026009.

Programs

  • Magma
    [Binomial(2*n-1, n) - Binomial(2*n-1, n+3): n in [1..30]]; // Vincenzo Librandi, Jun 21 2013
  • Maple
    a:= n-> binomial(2*n-1, n) -binomial(2*n-1, n+3): seq(a(n), n=1..27); #  Zerinvary Lajos, Dec 10 2007
  • Mathematica
    Table[Binomial[2 n - 1, n] - Binomial[2 n - 1, n + 3], {n, 1, 40}] (* Vincenzo Librandi, Jun 21 2013 *)
  • PARI
    a(n) = binomial(2*n-1, n) - binomial(2*n-1, n+3) \\ Charles R Greathouse IV, Oct 23 2023
  • Sage
    [binomial(2*n-1, n) - binomial(2*n-1, n+3) for n in (2..30)] # G. C. Greubel, Mar 19 2021

Formula

Expansion of (1+x^1*C^3)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
(n+3)*a(n) +(-7*n-9)*a(n-1) +2*(7*n-4)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jun 20 2013
From G. C. Greubel, Mar 19 2021: (Start)
G.f.: (1-x)*( 1 -4*x +2*x^2 - (1-2*x)*sqrt(1-4*x) )/(2*x^3).
E.g.f.: -1 + (exp(2*x)/x^2)*( x*(1+x)*BesselI(0, 2*x) - (2 +x +2*x^2)*Bessel(1, 2*x) ).
a(n) = C(n) + Sum_{j=0..n-2} C(n-j-2)*(C(j+3) - 2*C(j+2)), where C(n) are the Catalan numbers. (End)
a(n) = C(n+2) -3*C(n+1) +2*C(n) = 6*((n^2+1)/((n+2)*(n+3)))*C(n). - G. C. Greubel, Mar 22 2021

Extensions

Better description from Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001

A026020 a(n) = binomial(4n, n) - binomial(4n, n - 3).

Original entry on oeis.org

1, 4, 28, 219, 1804, 15314, 132572, 1163565, 10316924, 92195488, 829016968, 7492106505, 67991427828, 619193535380, 5655829748520, 51794730347745, 475390078267356, 4371917301657488, 40276635724273936, 371630891401943020, 3433826368544377520, 31768260456301092090
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

a(n) = T(4n, n), where T is the array defined in A026009.
Bisections are A026012 and A026016.
Cf. A004333, A005810.

Programs

  • Magma
    [Binomial(4*n, n) - Binomial(4*n, n-3): n in [0..20]]; // G. C. Greubel, Mar 22 2021
  • Maple
    A026020:= n-> binomial(4*n,n) - binomial(4*n,n-3); seq(A026020(n), n=0..20); # G. C. Greubel, Mar 22 2021
  • Mathematica
    Table[Binomial[4n, n] - Binomial[4n, n - 3], {n, 0, 19}] (* Alonso del Arte, Jun 06 2019 *)
  • PARI
    a(n) = binomial(4*n, n) - binomial(4*n, n-3) \\ Felix Fröhlich, Jun 06 2019
  • Sage
    [binomial(4*n, n) - binomial(4*n, n-3) for n in (0..20)] # G. C. Greubel, Mar 22 2021

Formula

G.f.: (g - 2)*(1 - g + g^2)*g/(3*g - 4) where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
a(n) = A005810(n) - A004333(n) for n > 2 - Felix Fröhlich, Jun 06 2019
a(n) ~ 13 * 2^(8*n+3/2) / (3^(3*n+7/2) * sqrt(Pi*n)). - Amiram Eldar, Sep 06 2025

A026019 a(n) = binomial(3*n,n) - binomial(3*n,n-3).

Original entry on oeis.org

1, 3, 15, 83, 483, 2898, 17748, 110295, 692967, 4390815, 28009215, 179652564, 1157534420, 7486680048, 48579667704, 316107403839, 2061920664351, 13478362911825, 88272020923485, 579081767982795, 3804622827123195, 25030721583818790, 164880542098818000, 1087305116074147620
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

a(n) = T(3n, n), where T is the array defined in A026009.
Cf. A004321, A005809.

Programs

  • Mathematica
    Table[Binomial[3n,n]-Binomial[3n,n-3],{n,0,20}] (* Harvey P. Dale, Jun 04 2016 *)
  • PARI
    a(n) = binomial(3*n,n) - binomial(3*n,n-3); \\ Michel Marcus, May 10 2020

Formula

G.f.: (1-2*g)*(g^2-g+1)/((3*g-1)*(g-1)^3) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
Conjecture: -2*(2*n+3)*(13*n-9)*(n+1)*a(n) + (499*n^3-7*n^2-120*n-54)*a(n-1) - 3*(3*n-5)*(37*n-24)*(3*n-4)*a(n-2) = 0. - R. J. Mathar, Jun 20 2013
From Amiram Eldar, Sep 06 2025: (Start)
a(n) = A005809(n) - A004321(n).
a(n) ~ 7 * 3^(3*n+1/2) / (4^(n+2) * sqrt(Pi*n)). (End)

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005

A061557 a(n) = (7*n+2)*C(n)/(n+2), where C(n) is the n-th Catalan number.

Original entry on oeis.org

3, 8, 23, 70, 222, 726, 2431, 8294, 28730, 100776, 357238, 1277788, 4605980, 16715250, 61020495, 223931910, 825632610, 3056887680, 11360977650, 42368413620, 158498860260, 594636663660, 2236748680998, 8433988655580
Offset: 1

Views

Author

Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001

Keywords

Comments

The identity a(n) = Sum_{k = 0..n} 3*(k-1)*C(k)*C(n-k)/(2*k-1) was verified using the Wilf-Zeilberger theory for hypergeometric sums. The sum arises in the enumeration of separable 1324-avoiding permutations: A026009(n) = a(n)/2 + 2*C(n-1) - 5*C(n)/2.
a(n) = 2*C(n+1) - C(n), with C(n) = A000108(n). - Ralf Stephan, Jan 13 2004

Crossrefs

Cf. A000108, A026009.
A000782 shifted left.
Previous Showing 11-18 of 18 results.

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

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