A001405 -id:A001405 - cp's OEIS frontend

A153585 Convolution triangle, A053121 * (A001405 * 0^(n-k)).

1, 0, 1, 1, 0, 2, 0, 2, 0, 3, 2, 0, 6, 0, 6, 0, 5, 0, 12, 0, 10, 5, 0, 18, 0, 30, 0, 20, 0, 14, 0, 42, 0, 60, 0, 35, 14, 0, 56, 0, 120, 0, 140, 0, 70, 0, 42, 0, 144, 0, 270, 0, 280, 0, 126

Offset: 0

Gary W. Adamson, Dec 28 2008

Row sums = A145974: (1, 1, 3, 5, 14, 27, 73,...).

			First few rows of the triangle =
1;
0, 1;
1, 0, 2;
0, 2, 0, 3;
2, 0, 6, 0, 6;
0, 5, 0, 12, 0, 10;
5, 0, 18, 0, 30, 0, 20;
0, 14, 0, 42, 0, 60, 0, 35;
14, 0, 56, 0, 120, 0, 140, 0, 70;
0, 42, 0, 144, 0, 270, 0, 280, 0, 126;
42, 0, 180, 0, 450, 0, 700, 0, 630, 0, 252;
...
Example: Row 4 = (2, 0, 6, 0, 6) = termwise products of (2, 0, 3, 0, 1) and (1, 1, 2, 3, 6).

Convolution triangle, A053121 * (A001405 * 0^(n-k)).
A053121 = the aerated Catalan triangle and (A001405 * 0^(n-k) = an
infinite lower triangular matrix with A001405 as the main diagonal and
the rest zeros.

A048627 Numbers m such that the maximal value of A001222(binomial(m,k)) and the central value A001222(A001405(m)) are identical.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 22, 23, 26, 27, 28, 29, 30, 39, 45, 46, 47, 51, 58, 59, 61, 62, 63, 86, 87, 93, 94, 95, 118, 119, 122, 123, 124, 125, 126, 147, 148, 158, 159, 178, 179, 187, 188, 189, 190, 214, 215, 221, 222, 236, 237, 238, 245, 246, 247, 248, 249, 253, 254

Offset: 1

Labos Elemer

nonn

Indexes of 0's in A048622. - Sean A. Irvine, Jun 24 2021

			For m=23, A001222 for binomial(23,k) = {0,1,2,3,4,4,5,5,6,6,6,6,6,6,6,6,5,5,4,4,3,2,1,0}, thus both the maximal and central values are 6, so 23 is a term.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A001222, A020731, A048622.

Select[Range[120], Function[n, ar = PrimeOmega[#] & /@ Binomial[n, Range[0, n/2]]; Max[ar] == ar[[-1]]]] (* Ivan Neretin, Sep 07 2015 *)

isok(m) = vecmax(apply(bigomega, vector(m+1, k, binomial(m,k-1)))) == bigomega(binomial(m, m\2)); \\ Michel Marcus, Jun 25 2021

A056201 Characteristic cube divisor (A056191) of central binomial coefficient (A001405).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 8, 27, 216, 8, 1, 1, 1, 27, 27, 1, 1, 1, 8, 1, 1, 1, 8, 1, 8, 216, 27, 1, 1, 1, 8, 27, 216, 8, 1, 1, 8, 8, 1, 8, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 8, 1, 8, 8, 1, 1, 1, 1, 8, 27, 216, 216, 27, 1, 8, 8, 1, 8, 1, 27, 216

Offset: 1

Labos Elemer, Aug 02 2000

nonn

			n=14, binomial(14,8) = 3432 = 2*2*2*3*11*13 so g(3432)=2, thus a(14)=8.

Cf. A056056-A056060, A056202.

Equals A056059^3.

a(n) = A056059(n)^3 = g^3 and binomial(n, floor(n/2)) = a(n)^3 * L^2 * A056060(n), where L = A056056(n)/A056059(n).

A056647 a(n) = A056623(A001405(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 9, 36, 1, 4, 4, 1, 9, 9, 1, 4, 1, 4, 4, 1, 1, 4, 100, 25, 100, 25, 9, 144, 9, 9, 1, 4, 25, 100, 100, 25, 9, 36, 4, 1, 4, 1, 25, 400, 225, 900, 1764, 441, 196, 49, 49, 784, 4, 1, 1, 16, 1, 16, 16, 4, 441, 441, 49, 196, 49, 196, 36, 9, 1, 4, 4, 1, 100, 25, 1225

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name "Largest unitary square divisor of central binomial coefficient" was incorrect. See A376553 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A056623(binomial(28,14)) = A056623(40116600) = 25.

Cf. A001405, A008833, A055229, A056056, A056057, A056059, A056623, A376553.

a(n) = A008833(A001405(n))/A055229(A001405(n))^2 = A056057(n)/A056059(n)^2.

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

A056673 Number of unitary and squarefree divisors of binomial(n, floor(n/2)). Also the number of divisors of the powerfree part of A001405(n), A056060(n).

Original entry on oeis.org

1, 2, 2, 4, 4, 2, 4, 8, 4, 2, 16, 8, 8, 8, 8, 16, 32, 16, 32, 16, 32, 32, 64, 32, 16, 16, 8, 8, 32, 32, 64, 128, 128, 64, 256, 128, 128, 128, 512, 256, 512, 512, 512, 512, 64, 64, 256, 128, 128, 128, 128, 128, 256, 256, 2048, 2048, 4096, 4096, 2048, 2048, 2048, 2048

Offset: 1

Labos Elemer, Aug 10 2000

nonn

			n = 14: binomial(15,7) = 3432 = 2*2*2*3*11*13, which has 32 divisors. Of those divisors, 16 are unitary: {1, 3, 8, 11, 13, 24, 33, 39, 88, 104, 143, 264, 312, 429, 1144, 3432}; 16 are squarefree: {1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858}. Only 8 of the divisors belong to both classes: {1, 3, 11, 13, 33, 39, 143, 429}. Thus, a(14) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001405, A007913, A055229, A055231, A056060, A056671.

Table[With[{m = Binomial[n, Floor[n/2]]}, DivisorSum[m, 1 &, And[CoprimeQ[#, m/#], SquareFreeQ@ #] &]], {n, 62}] (* Michael De Vlieger, Sep 05 2017 *)
f[p_, e_] := If[e == 1, 2, 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[ Binomial[n, Floor[n/2]]]); Array[a, 60] (* Amiram Eldar, Sep 06 2020 *)

a(n) = my(b=binomial(n, n\2)); sumdiv(b, d, issquarefree(d) && (gcd(d, b/d) == 1)); \\ Michel Marcus, Sep 05 2017

a(n) = A000005(A055231(x)) = A000005(A007913(x)/A055229(x)), where x = A001405(n) = binomial(n, floor(n/2)).

a(n) = A056671(A001405(n)). - Amiram Eldar, Sep 06 2020

A062791 Moebius transform of A001405 (binomial(n, floor(n/2))).

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 34, 64, 123, 241, 461, 900, 1715, 3396, 6423, 12800, 24309, 48477, 92377, 184500, 352679, 704969, 1352077, 2703168, 5200290, 10398883, 20058174, 40113164, 77558759, 155110827, 300540194, 601067520, 1166802646, 2333581909, 4537567606

Offset: 1

Labos Elemer, Jul 19 2001

nonn

			For n = 7, binomial(7,3) = 35, A001405(7/d) = {binomial(7,3), binomial(1,0)} = {35, 1}, mu(d) = {1, -1}, the sum is a(7) = 35 - 1 = 34.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A001405, A007727, A062798.

with(numtheory):
a:= n-> add(binomial(d, iquo(d, 2))*mobius(n/d), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 09 2017

a[n_] := DivisorSum[n, Binomial[#, Floor[#/2]] * MoebiusMu[n/#] &]; Array[a, 35] (* Amiram Eldar, May 28 2025 *)

a(n) = sumdiv(n, d, binomial(d, d\2) * moebius(n/d)); \\ Amiram Eldar, May 28 2025

a(n) = Sum_{d|n} A001405(n/d)*mu(d).

Offset corrected by Eric Rowland, Jul 09 2017

A062798 Inverse Moebius transform of A001405 (binomial(n, floor(n/2))).

Original entry on oeis.org

1, 3, 4, 9, 11, 26, 36, 79, 130, 265, 463, 956, 1717, 3470, 6449, 12949, 24311, 48772, 92379, 185027, 352755, 705897, 1352079, 2705182, 5200311, 10402319, 20058430, 40120076, 77558761, 155124243, 300540196, 601093339, 1166803576, 2333630533

Offset: 1

Labos Elemer, Jul 19 2001

nonn

			For n = 9, divisors = {1,3,9}, a(9) = binomial(1, 0) + binomial(3, 1) + binomial(9, 4) = 1 + 3 + 126 = 130.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A001405, A062791, A072929.

a[n_] := DivisorSum[n, Binomial[#, Floor[#/2]]&]; Array[a, 35] (* Amiram Eldar, May 28 2025 *)

a(n) = sumdiv(n, d, binomial(d, d\2)); \\ Amiram Eldar, May 28 2025

a(n) = Sum_{d|n} binomial(d, floor(d/2)).

Offset corrected by Amiram Eldar, May 28 2025

A081394 a(n) is the smallest k such that number of non-unitary prime divisors of central binomial coefficient, A001405(k) = C(k, floor(k/2)) equals n.

Original entry on oeis.org

1, 6, 10, 27, 96, 147, 363, 627, 959, 1547, 1919, 2641, 2645, 3339, 6241, 6909, 6913, 6943, 6923, 6937, 16405, 19981, 24325, 31675, 31679, 35329, 36959, 36963, 38915, 38927, 73563, 39729, 73577, 80095, 87205, 87309, 95035, 123307, 123305, 123369, 123367, 174239, 185915, 186361, 186369, 186373, 186381

Offset: 0

Labos Elemer, Mar 27 2003

nonn

			n=8: a(8)=959, C(959,479) has 8 non-unitary prime divisors: {2,3,5,7,11,13,23,29} and 959 is the smallest.

Cf. A081387, A081393, A081395, A056175.

seq[len_, kmax_] := Module[{s = Table[0, {len}], k = 1, c = 0, i}, While[c < len && k < kmax, i = Count[FactorInteger[Binomial[k, Floor[k/2]]][[;; , 2]], ?(# > 1 &)] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = k]; k++]; TakeWhile[s, # > 0 &]]; seq[20, 10^4] (* _Amiram Eldar, May 15 2023 *)

a(n) = Min{k; A056175(k) = n}.

a(9)-a(19) from Michel Marcus, Sep 01 2019

a(20)-a(46) from Amiram Eldar, May 15 2023

A113409 A transform of the central binomial coefficients A001405.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 21, 39, 74, 141, 271, 521, 1004, 1939, 3756, 7291, 14176, 27599, 53805, 105031, 205268, 401573, 786328, 1541037, 3022528, 5932657, 11652617, 22901865, 45037432, 88616807, 174454943, 343606183, 677074350, 1334744305

Offset: 0

Paul Barry, Oct 28 2005

Row sums of A113408.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Table[Sum[Binomial[n - k, k]*Binomial[k, Floor[k/2]], {k, 0, Floor[n/2]}], {n,0,50}] (* G. C. Greubel, Mar 09 2017 *)

for(n=0,25, print1(sum(k=0,floor(n/2), binomial(n-k,k)*binomial(k,floor(k/2))), ", ")) \\ G. C. Greubel, Mar 09 2017

G.f.: (1-xc(x^2))/(1-x^2-x^4c(x^4)), where c(x) is the g.f. of A000108.
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*C(k, floor(k/2)).
a(n) = Sum_{k=0..n} C((n+k)/2, k)*C(floor((n-k)/2), floor((n-k)/4)).
Conjecture: (n+2)*a(n)-2*(n+1)*a(n-1) +(n-4)*a(n-2) +2*a(n-3) +4*(2-n)*a(n-4)=0. - R. J. Mathar, Nov 07 2012
a(n) ~ 2^(n + 3/2) / sqrt(3*Pi*n). - Vaclav Kotesovec, Nov 27 2017

A129383 Expansion of g(x) - x*g(x^2), where g(x) is the g.f. of A001405.

Original entry on oeis.org

1, 0, 2, 2, 6, 8, 20, 32, 70, 120, 252, 452, 924, 1696, 3432, 6400, 12870, 24240, 48620, 92252, 184756, 352464, 705432, 1351616, 2704156, 5199376, 10400600, 20056584, 40116600, 77555328, 155117520, 300533760, 601080390, 1166790240

Offset: 0

Paul Barry, Apr 12 2007

Partial sums are A129384.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001405, A129384.

A129383:= func< n | Binomial(n,Floor(n/2)) - (n mod 2)*Binomial(Floor((n-1)/2),Floor((n-1)/4))  >;
[A129383(n): n in [0..40]]; // G. C. Greubel, Feb 03 2024

A129383[n_]:= With[{B=Binomial,F=Floor}, B[n,F[n/2]] - Mod[n,2]*B[(n- 1)/2, F[(n-1)/4]]];
Table[A129383[n], {n,0,40}] (* G. C. Greubel, Feb 03 2024 *)

def A129383(n): return binomial(n,n//2) - (n%2)*binomial((n-1)/2,(n-1)//4)
[A129383(n) for n in range(41)] # G. C. Greubel, Feb 03 2024

G.f.: 2/(1-2*x+sqrt(1-4*x^2)) - 2*x/(1-2*x^2+sqrt(1-4*x^4)).

a(n) = binomial(n,floor(n/2)) - (1/2)*(1-(-1)^n)*binomial((n-1)/2, floor((n-1)/4)).

cp's OEIS Frontend

A153585 Convolution triangle, A053121 * (A001405 * 0^(n-k)).

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A048627 Numbers m such that the maximal value of A001222(binomial(m,k)) and the central value A001222(A001405(m)) are identical.

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A056201 Characteristic cube divisor (A056191) of central binomial coefficient (A001405).

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A056647 a(n) = A056623(A001405(n)).

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A056673 Number of unitary and squarefree divisors of binomial(n, floor(n/2)). Also the number of divisors of the powerfree part of A001405(n), A056060(n).

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A062791 Moebius transform of A001405 (binomial(n, floor(n/2))).

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A062798 Inverse Moebius transform of A001405 (binomial(n, floor(n/2))).

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A081394 a(n) is the smallest k such that number of non-unitary prime divisors of central binomial coefficient, A001405(k) = C(k, floor(k/2)) equals n.

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