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.

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A187442 A trisection of A001405 (central binomial coefficients): binomial(3*n,floor(3*n/2)), n>=0.

Original entry on oeis.org

1, 3, 20, 126, 924, 6435, 48620, 352716, 2704156, 20058300, 155117520, 1166803110, 9075135300, 68923264410, 538257874440, 4116715363800, 32247603683100, 247959266474052, 1946939425648112, 15033633249770520, 118264581564861424
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For the trisection of sequences see a comment and a reference under A187357.

Crossrefs

A187443 binomial(3*n+1,floor((3*n+1)/2)),
A187444 binomial(3*n+2,floor((3*n+2)/2))/2.

Programs

  • Mathematica
    Table[Binomial[3n,Floor[(3n)/2]],{n,0,20}] (* Harvey P. Dale, Dec 23 2012 *)

Formula

a(n) = binomial(3*n,floor(3*n/2)), n>=0.
O.g.f.: G0(x^2) + 3*x*G2(x^2) with G0(x) and G2(x) the o.g.f.s of A187363 and A187365, respectively.

A187443 A trisection of A001405 (central binomial coefficients): binomial(3n+1,floor((3n+1)/2)), n>=0.

Original entry on oeis.org

1, 6, 35, 252, 1716, 12870, 92378, 705432, 5200300, 40116600, 300540195, 2333606220, 17672631900, 137846528820, 1052049481860, 8233430727600, 63205303218876, 495918532948104, 3824345300380220, 30067266499541040, 232714176627630544, 1832624140942590534, 14226520737620288370, 112186277816662845432, 873065282167813104916
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For trisection of sequences see a comment and a reference under A187357.

Crossrefs

Cf. A187442: binomial(3n,floor(3n/2)), A187444: binomial(3n+2,floor((3n+2)/2))/2.
Cf. A187357, A187364, A187365.
Cf. A001405.

Programs

  • Mathematica
    Table[Binomial[3n+1,Floor[(3n+1)/2]],{n,0,30}] (* Harvey P. Dale, Jan 13 2021 *)

Formula

a(n) = binomial(3*n+1,floor((3*n+1)/2)), n>=0.
O.g.f.: 3!*x*G2(x^2) + G1(x^2), with G2(x) and G1(x) the o.g.f.s of A187365 and A187364, respectively.

Extensions

Corrected and extended by Harvey P. Dale, Jan 13 2021

A187444 A trisection of A001405 (central binomial coefficients): binomial(3n+2,floor((3n+2)/2))/2, n>=0.

Original entry on oeis.org

1, 5, 35, 231, 1716, 12155, 92378, 676039, 5200300, 38779380, 300540195, 2268783825, 17672631900, 134564468610, 1052049481860, 8061900920775, 63205303218876, 486734856412028, 3824345300380220, 29566145391215356, 232714176627630544, 1804857108504066435
Offset: 0

Views

Author

Wolfdieter Lang, Mar 10 2011

Keywords

Comments

For the trisection of sequences see a comment and a reference under A187357.

Crossrefs

Cf. A187442: binomial(3*n,floor(3*n/2)), A187443: binomial(3*n+1,floor((3*n+1)/2)).

Programs

  • PARI
    vector(30, n, n--; binomial(3*n+2,(3*n+2)\2)/2) \\ Michel Marcus, Jun 11 2015

Formula

a(n) = binomial(3*n+2,floor((3*n+2)/2))/2, n>=0.
O.g.f.: G1(x^2) + x*G2(x^2), with G1(x) and G2(x) the o.g.f.s of A187364 and A187366, respectively.

A048244 a(n) = A048106(A001405(n)).

Original entry on oeis.org

1, 2, 2, 4, 4, 2, 4, 8, 4, -2, 16, 8, 8, 0, 8, 16, 32, 16, 32, 16, 32, 0, 64, 32, -16, -64, -160, -256, -128, -224, 64, 128, 0, -256, 256, -128, -128, -512, 512, -256, 512, 0, 512, 0, -2048, -2816, -256, -1408, -1408, -2560, -2560, -4096, -1024, -1792, 2048
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

The terms indicate whether more, equal or fewer unitary than non-unitary divisors of the central binomial coefficient exists.

Examples

			For n = 54, binomial(54,27) has 3840 divisors of which 1024 are unitary and 2816 are not. The difference is -1792, so a(54) = -1792.

Links

Crossrefs

Cf. A001405, A048106.

Programs

  • Mathematica
    a[n_] := Module[{b = Binomial[n, Floor[n/2]]}, 2^(PrimeNu[b] + 1) - DivisorSigma[0, b]]; Array[a, 60] (* Amiram Eldar, Oct 05 2024 *)
  • PARI
    a048106(n) = (2^(1+omega(n)) - numdiv(n));
a(n) = a048106(binomial(n, n\2)); \\ Michel Marcus, May 14 2018

Formula

a(n) = A034444(A001405(n)) - A048105(A001405(n)).

A048682 a(n) is the difference between maximal and central squarefree kernel numbers dividing values of {binomial(n,k)} or A001405(n), respectively.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 0, 0, 0, 168, 0, 0, 0, 2145, 2860, 0, 0, 0, 0, 33592, 117572, 470288, 0, 0, 297160, 4791705, 2674440, 12900240, 28134060, 10835415, 0, 0, 0, 2074316640, 0, 0, 2524661700, 31810737420, 39384722520, 0, 0, 0, 82334307276, 1235014609140
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			n=10: the squarefree kernels are {1, 10, 15, 30, 210, 42, 210, 30, 15, 10, 1}. The maximal value is 210 and the central one is 42. Thus a(10) = 210 - 42 = 168.

Crossrefs

Analogous cases for A001222, A000005 as applied to {binomial(n, k)} are given in A048623, A020740.

A054442 Second convolution of A001405 (central binomial numbers).

Original entry on oeis.org

1, 3, 9, 22, 54, 123, 281, 618, 1362, 2934, 6330, 13452, 28620, 60243, 126921, 265282, 554874, 1153506, 2399390, 4966740, 10286196, 21219038, 43790154, 90076452, 185353204, 380364108, 780786516, 1599015192, 3275589144
Offset: 0

Views

Author

Wolfdieter Lang, Mar 27 2000

Keywords

Crossrefs

Cf. A000108, A001405, A054336, A045621.

Formula

a(2*k)= -3*2^(2*k+1)+binomial(2*(k+1), k+1)*(4*k+7)/2, a(2*k+1)= -3*4^(k+1)+binomial(2*(k+2), k+2)*(2*k+5)/2, k >= 0.
a(n)= A054336(n+2, 2) (third column of convolution triangle). G.f.: (1/(1-x-x^2*c(x^2)))^3, with c(x) the g.f. for the Catalan numbers A000108.

A054443 Third convolution of A001405 (central binomial numbers).

Original entry on oeis.org

1, 4, 14, 40, 109, 276, 682, 1624, 3810, 8744, 19868, 44496, 98941, 217780, 476786, 1036024, 2241814, 4823160, 10342180, 22076080, 46994386, 99673224, 210923364, 445000560, 937051684, 1968204496, 4127285688, 8636324768, 18045851165, 37638105588, 78404375362
Offset: 0

Views

Author

Wolfdieter Lang, Mar 27 2000

Keywords

Links

Crossrefs

Cf. A000108, A001405, A054336, A054442.

Programs

  • PARI
    {a(n)=local(k);if(n<0, 0, k=n\2; if(n%2, (k+4)*4^(k+2)-(k+3)*binomial(2*(k+3),k+3), (2*k+7)*4^(k+1)-binomial(2*(k+2),k+2)*(4*k+9)/2 ))}

Formula

a(2*k) = (2*k+7)*4^(k+1)-binomial(2*(k+2), k+2)*(4*k+9)/2, a(2*k+1) = (k+4)*4^(k+2)-(k+3)*binomial(2*(k+3), k+3), k >= 0.
a(n) = A054336(n+3, 3) (fourth column of convolution triangle). G.f.: (1/(1-x-x^2*c(x^2)))^4, with c(x) the g.f. for the Catalan numbers A000108.
G.f.: (c(x/(2x-1))/(1-2x))^4. - Michael Somos, Jul 31 2005

A056202 Central binomial coefficient A001405(n) divided by its characteristic cube divisor A056201(n).

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716, 429, 6435, 12870, 24310, 48620, 92378, 184756, 352716, 88179, 1352078, 2704156, 5200300, 1300075, 742900, 185725, 9694845, 155117520, 300540195, 601080390, 43214930, 86429860
Offset: 1

Views

Author

Labos Elemer, Aug 02 2000

Keywords

Examples

			n=14, binomial(14,7) = 3432 and A056059(14) = 2, thus a(14) = 3432/(2*2*2) = 429.

Crossrefs

Cf. A001405, A056056-A056060, A056201.

Formula

a(n) = A001405(n)/A056059(n)^3 = binomial(n, floor(n/2))/A056059(n)^3 = A001405(n)/A056201(n).

A056670 Largest non-unitary prime factor of A001405(n) = binomial(n, floor(n/2)), or 1 if no such prime exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 2, 2, 3, 3, 1, 2, 1, 2, 2, 2, 1, 2, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 3, 3, 2, 2, 2, 2, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 2, 2, 2, 2, 2, 2, 2, 2, 7, 7, 7, 7, 7, 7, 3, 3, 1, 2, 2, 2, 5, 5, 7, 7, 7, 7, 7, 7, 3, 3, 5, 5, 5, 5, 3, 3, 7, 7, 7, 7, 7, 7, 5, 5, 3, 3, 3, 3, 3, 3, 3
Offset: 1

Views

Author

Labos Elemer, Aug 10 2000

Keywords

Comments

The largest prime divisor of A056057(n), the largest square divisor of binomial(n, floor(n/2)), or 1 if no such prime exists.

Examples

			For n = 28: binomial(28,14) = 2*2*2*3*3*3*5*5*17*19*23, so a(28) = 5.
For n = 342: binomial(342,171) = 32*F, where F is squarefree, so a(341) = 2.

Links

Crossrefs

Cf. A001405, A006530, A056057, A056175.

Programs

  • Mathematica
    a[n_] := Module[{f = Select[FactorInteger[Binomial[n, Floor[n/2]]], Last[#] > 1 &]}, If[f == {}, 1, f[[-1, 1]]]]; Array[a, 100] (* Amiram Eldar, Oct 05 2024 *)

Formula

a(n) = A006530(A056057(n)).

A067366 Numbers n such that 1+C(n,floor(n/2)) is prime, where C(n,floor(n/2)) = A001405(n) is a central binomial coefficient.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 11, 14, 24, 27, 37, 38, 44, 47, 76, 89, 92, 97, 123, 124, 136, 143, 144, 168, 243, 297, 332, 368, 393, 417, 428, 443, 680, 696, 697, 817, 890, 981, 1034, 1053, 1201, 1349, 1384, 1455, 1537, 1589, 1634, 1705, 2077, 2632, 2635, 2762, 3197
Offset: 1

Views

Author

Dean Hickerson, Jan 20 2002

Keywords

Comments

The corresponding primes are in A066749.

Crossrefs

Cf. A001405, A066749.

Programs

  • Mathematica
    For[ n=0, True, n++, If[ PrimeQ[ Binomial[ n, Floor[ n/2 ] ]+1 ], Print[ n ] ] ]
Previous Showing 21-30 of 434 results. Next

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