A163455 -id:A163455 - cp's OEIS frontend

A262977 a(n) = binomial(4*n-1,n).

1, 3, 21, 165, 1365, 11628, 100947, 888030, 7888725, 70607460, 635745396, 5752004349, 52251400851, 476260169700, 4353548972850, 39895566894540, 366395202809685, 3371363686069236, 31074067324187580, 286845713747883300, 2651487106659130740, 24539426037817994160

Offset: 0

Vladimir Kruchinin, Oct 06 2015

From Gus Wiseman, Sep 28 2022: (Start)
Also the number of integer compositions of 4n with alternating sum 2n, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These compositions are ranked by A348614. The a(12) = 21 compositions are:
  (6,2)  (1,2,5)  (1,1,5,1)  (1,1,1,1,4)
         (2,2,4)  (2,1,4,1)  (1,1,2,1,3)
         (3,2,3)  (3,1,3,1)  (1,1,3,1,2)
         (4,2,2)  (4,1,2,1)  (1,1,4,1,1)
         (5,2,1)  (5,1,1,1)  (2,1,1,1,3)
                             (2,1,2,1,2)
                             (2,1,3,1,1)
                             (3,1,1,1,2)
                             (3,1,2,1,1)
                             (4,1,1,1,1)
The following pertain to this interpretation:
- The case of partitions is A000712, reverse A006330.
- Allowing any alternating sum gives A013777 (compositions of 4n).
- A011782 counts compositions of n.
- A034871 counts compositions of 2n with alternating sum 2k.
- A097805 counts compositions by alternating (or reverse-alternating) sum.
- A103919 counts partitions by sum and alternating sum (reverse: A344612).
- A345197 counts compositions by length and alternating sum.
(End)

V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Cf. A006632, A002293, A004331, A005810, A052203, A224274, A257633.
Cf. A000346, A000984, A001700, A002458, A025047, A058622, A081294, A294175.
Cf. A088218, A163455, A165817.

[Binomial(4*n-1,n): n in [0..20]]; // Vincenzo Librandi, Oct 06 2015

Table[Binomial[4 n - 1, n], {n, 0, 40}] (* Vincenzo Librandi, Oct 06 2015 *)

B(x):=sum(binomial(4*n-1,n-1)*3/(4*n-1)*x^n,n,1,30);
taylor(x*diff(B(x),x,1)/B(x),x,0,20);

a(n) = binomial(4*n-1,n); \\ Michel Marcus, Oct 06 2015

G.f.: A(x)=x*B'(x)/B(x), where B(x) if g.f. of A006632.
a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*binomial(3*n,k)).
a(n) = 3*A224274(n), for n > 0. - Michel Marcus, Oct 12 2015
From Peter Bala, Nov 04 2015: (Start)
The o.g.f. equals f(x)/g(x), where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A005810 (k = 0), A052203 (k = 1), A257633 (k = 2), A224274 (k = 3) and A004331 (k = 4). (End)
a(n) = [x^n] 1/(1 - x)^(3*n). - Ilya Gutkovskiy, Oct 03 2017
a(n) = A071919(3n-1,n+1) = A097805(4n,n+1). - Gus Wiseman, Sep 28 2022
From Peter Bala, Feb 14 2024: (Start)
a(n) = (-1)^n * binomial(-3*n, n).
a(n) = hypergeom([1 - 3*n, -n], [1], 1).
The g.f. A(x) satisfies A(x/(1 + x)^4) = 1/(1 - 3*x). (End)
a(n) = Sum_{k = 0..n} binomial(2*n+k-1, k)*binomial(2*n-k-1, n-k). - Peter Bala, Sep 16 2024
G.f.: 1/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A163456 a(n) = binomial(5*n,n)/5.

Original entry on oeis.org

1, 9, 91, 969, 10626, 118755, 1344904, 15380937, 177232627, 2054455634, 23930713170, 279871768995, 3284214703056, 38650751381832, 456002537343216, 5391644226101705, 63871405575418665, 757929628541719755

Offset: 1

Zak Seidov, Jul 28 2009

For prime p, a(p) == 1 (mod p). - Gary Detlefs, Aug 03 2013
In fact, a(p) == 1 (mod p^3) for prime p >= 5. See Mestrovic, Section 3. - Peter Bala, Oct 09 2015
From Robert Israel, Jul 12 2016: (Start)
a(p+1) == 5 (mod p) for primes p >= 5.
a(p^(k+1)) == a(p^k) mod p^(3(k+1)) for primes p >= 5. (End)

Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 2nd ed. 1994.

Robert Israel, Table of n, a(n) for n = 1..920
Romeo Meštrović, Lucas’ theorem: its generalizations, extensions and applications (1878-2014), arXiv:1409.3820v1 [math.NT], 2014.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Cf. A000108, A000245, A001764, A002293, A025174, A118970, A163455, A224274, A227726.

seq(binomial(5*n,n)/5, n=1..20); # Robert Israel, Jul 12 2016

Array[Binomial[5 #, #]/5 &, {18}] (* Michael De Vlieger, Oct 09 2015 *)

a(n) = binomial(5*n,n)/5 \\ Altug Alkan, Oct 09 2015

a(n) = (5*n-1)!/(4*n!*(4*n-1)!) = A001449(n)/5 = A163455(n)/4.
a(n) = binomial(5*n,n)/5. - Gary Detlefs, Aug 03 2013
From Peter Bala, Oct 08 2015: (Start)
a(n) = (1/3)*[x^n] (C(x)^3)^n, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A224274.
exp( 3*Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 18*x^2 + 136*x^3 + ... is the o.g.f. for A118970. (End)
From Peter Bala,Jul 12 2016: (Start)
a(n) = 1/6*[x^n] (1 + x)/(1 - x)^(4*n + 1).
a(n) = 1/6*[x^n] ( 1/C(-x)^6 )^n. Cf. A227726. (End)
a(n) ~ 2^(-8*n-3/2)*5^(5*n-1/2)*n^(-1/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016
From Robert Israel, Jul 12 2016: (Start)
G.f.: x*hypergeom([1, 6/5, 7/5, 8/5, 9/5], [5/4, 3/2, 7/4, 2], (3125/256)*x).
a(n) = 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)/(8*n*(4*n-3)*(2*n-1)*(4*n-1)). (End)
O.g.f.: f(x)/(1 - 4*f(x)), where f(x) = series reversion (x/(1 + x)^5) = x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ... is the o.g.f. of A002294 with the initial term omitted. Cf. A025174. - Peter Bala, Feb 03 2022
Right-hand side of the identities (1/4)*Sum_{k = 0..n} (-1)^(n+k)*C(x*n,n-k)*C((x+4)*n+k-1,k) = C(5*n,n)/5 and (1/5)*Sum_{k = 0..n} (-1)^k*C(x*n,n-k)*C((x-5)*n+k-1,k) = C(5*n,n)/5, both valid for n >= 1 and x arbitrary. - Peter Bala, Feb 28 2022
Right-hand side of the identity (1/4)*Sum_{k = 0..2*n} (-1)^k*binomial(6*n-k-1,2*n-k)*binomial(4*n+k-1,k) = binomial(5*n,n)/5, for n >= 1. - Peter Bala, Mar 09 2022
a(n) = (1/2)* [x*n] F(x)^(2*n) = [x^n] G(x)^n for n >= 1, where F(x) = Sum_{k >= 0} 1/(2*k + 1)*binomial(3*k,k)*x^k is the o.g.f. of A001764 and G(x) = Sum_{k >= 0} 1/(3*k + 1)*binomial(4*k,k)*x^k is the o.g.f. of A002293 (apply Concrete Mathematics, equation 5.60, p. 201). - Peter Bala, Apr 26 2023

Renamed by Peter Bala, Oct 08 2015

A371739 a(n) = Sum_{k=0..n} binomial(5*n,k).

Original entry on oeis.org

1, 6, 56, 576, 6196, 68406, 768212, 8731848, 100146724, 1156626990, 13432735556, 156713948672, 1835237017324, 21560768699762, 253994850228896, 2999267652451776, 35490014668470052, 420718526924212654, 4995548847105422048, 59402743684137281920

Offset: 0

Seiichi Manyama, Apr 05 2024

Cf. A032443, A066380, A066381.

Cf. A002294, A163455, A386699.

Table[32^n - Binomial[5*n, 1+n] * Hypergeometric2F1[1, 1 - 4*n, 2+n, -1], {n, 0, 20}] (* Vaclav Kotesovec, Apr 05 2024 *)

PARI
```
a(n) = sum(k=0, n, binomial(5*n, k));
```

a(n) = [x^n] 1/((1-2*x) * (1-x)^(4*n)).
a(n) ~ 5^(5*n + 1/2) / (3*sqrt(Pi*n) * 2^(8*n - 1/2)). - Vaclav Kotesovec, Apr 05 2024
a(n) = Sum_{k=0..floor(n/2)} binomial(5*n+1,n-2*k). - Seiichi Manyama, Apr 09 2024
a(n) = binomial(1+5*n, n)*hypergeom([1, (1-n)/2, -n/2], [1+2*n, 3/2+2*n], 1). - Stefano Spezia, Apr 09 2024
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+k-1,k). - Seiichi Manyama, Jul 30 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 08 2025
G.f.: g/((2-g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 12 2025
G.f.: 1/(1 - x*g^4*(10-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

A386699 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n,k).

Original entry on oeis.org

1, 7, 69, 733, 8061, 90462, 1028871, 11814376, 136643085, 1589311381, 18569375114, 217773347502, 2561944357311, 30219704365104, 357278540928168, 4232449819704768, 50227362114232109, 596988743410929087, 7105534815529752831, 84678089652554263155, 1010268312800732117946

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A000244, A100192, A385004, A385498.
Cf. A163455, A371739, A386702.
Cf. A002294.

Table[(243/16)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, -1/2]), {n,0,25}] (* Vaclav Kotesovec, Jul 30 2025 *)

a(n) = sum(k=0, n, 2^(n-k)*binomial(5*n, k));

a(n) = [x^n] 1/((1-3*x) * (1-x)^(4*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 128*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(3052*n^4 - 15114*n^3 + 26432*n^2 - 18693*n + 4131)*a(n) = 8*(42807352*n^8 - 285737492*n^7 + 758983420*n^6 - 1002945218*n^5 + 644348866*n^4 - 111879380*n^3 - 84004497*n^2 + 44187381*n - 5806080)*a(n-1) - 1215*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(3052*n^4 - 2906*n^3 - 598*n^2 + 1037*n - 192)*a(n-2).
a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi*n) * 2^(8*n + 1/2)). (End)
G.f.: g/((3-2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^4*(15-8*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

A386702 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n,k).

Original entry on oeis.org

1, 2, 24, 248, 2676, 29562, 331956, 3771896, 43242660, 499215146, 5795429764, 67587697872, 791232339756, 9292673328174, 109440405341088, 1291977861163968, 15284200451058724, 181147979395807002, 2150493166839159936, 25567085678133719880, 304368033788893315896

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A386700, A386701.

Cf. A163455, A371739, A386699.

Table[(-32/81)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

a(n) = sum(k=0, n, (-3)^(n-k)*binomial(5*n, k));

a(n) = [x^n] 1/((1+2*x) * (1-x)^(4*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 648*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(37331*n^4 - 227972*n^3 + 518701*n^2 - 521044*n + 194928)*a(n) = (9143593823*n^8 - 74277961298*n^7 + 253684378280*n^6 - 473415527402*n^5 + 524935655069*n^4 - 351762123620*n^3 + 137998180332*n^2 - 28677229776*n + 2380855680)*a(n-1) + 160*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(37331*n^4 - 78648*n^3 + 58771*n^2 - 18234*n + 1944)*a(n-2).
a(n) ~ 5^(5*n + 1/2) / (7 * sqrt(Pi*n) * 2^(8*n - 1/2)). (End)
G.f.: g/((-2+3*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^4*(-10+12*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

A318593 Number of n-member subsets of [5*n] whose elements sum to a multiple of five.

Original entry on oeis.org

1, 1, 9, 91, 969, 10630, 118755, 1344904, 15380937, 177232627, 2054455670, 23930713170, 279871768995, 3284214703056, 38650751381832, 456002537343580, 5391644226101705, 63871405575418665, 757929628541719755, 9007607943130625829, 107196674080761940470

Offset: 0

Alois P. Heinz, Aug 29 2018

nonn

			a(2) = 9: {1,4}, {1,9}, {2,3}, {2,8}, {3,7}, {4,6}, {5,10}, {6,9}, {7,8}.

Alois P. Heinz, Table of n, a(n) for n = 0..922

Column k=5 of A318557.

Cf. A163455, A163456.

b:= proc(n, s, m, t) option remember; `if`(n=0, `if`(s=0 and t=0, 1, 0),
      b(n-1, s, m, t)+`if`(t=0, 0, b(n-1, irem(s+n, m), m, t-1)))
    end:
a:= n-> b(5*n, 0, 5, n):
seq(a(n), n=0..27);

a(n) = floor(A163456(n)) + [n mod 5 = 0]*A163455(n/5), with A163456(n) = binomial(5*n,n)/5 and A163455(n) = binomial(5*n-1,n) where [] is an Iverson bracket. - Georg Fischer, Mar 23 2019

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A262977 a(n) = binomial(4*n-1,n).

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A163456 a(n) = binomial(5*n,n)/5.

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A371739 a(n) = Sum_{k=0..n} binomial(5*n,k).

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A386699 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n,k).

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A386702 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n,k).

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A318593 Number of n-member subsets of [5*n] whose elements sum to a multiple of five.

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