A052203 -id:A052203 - cp's OEIS frontend

A119304 Triangle read by rows: T(n,k) = binomial(4n-k,n-k), 0 <= k <= n.

1, 4, 1, 28, 7, 1, 220, 55, 10, 1, 1820, 455, 91, 13, 1, 15504, 3876, 816, 136, 16, 1, 134596, 33649, 7315, 1330, 190, 19, 1, 1184040, 296010, 65780, 12650, 2024, 253, 22, 1, 10518300, 2629575, 593775, 118755, 20475, 2925, 325, 25, 1, 94143280, 23535820

Offset: 0

Paul Barry, May 13 2006

			Triangle begins
       1;
       4,     1;
      28,     7,    1;
     220,    55,   10,    1;
    1820,   455,   91,   13,   1;
   15504,  3876,  816,  136,  16,  1;
  134596, 33649, 7315, 1330, 190, 19, 1;

Indranil Ghosh, Rows 0..125, flattened

Rows sums are A052203. First column is A005810. Inverse of A119305.

Cf. A092392, A119301.

Flatten[Table[Binomial[4n-k,n-k],{n,0,9},{k,0,n}]] (* Indranil Ghosh, Feb 26 2017 *)

tabl(nn) = {for (n=0,nn,for (k=0,n,print1(binomial(4*n-k,n-k),", ");); print(););} \\ Indranil Ghosh, Feb 26 2017

from sympy import binomial
i=0
for n in range(12):
    for k in range(n+1):
        print(str(i)+" "+str(binomial(4*n-k,n-k)))
        i+=1 # Indranil Ghosh, Feb 26 2017

Riordan array (1/(1-4f(x)),f(x)) where f(x)(1-f(x))^3 = x.
From Peter Bala, Jun 04 2024: (Start)
'Horizontal' recurrence equation: T(n, 0) = binomial(4*n,n) and for k >= 1, T(n, k) = Sum_{i = 1..n+1-k} i*(i+1)/2 * T(n-1, k-2+i).
T(n, k) = Sum_{j = 0..n} binomial(n+j-1, j)*binomial(3*n-k-j, 2*n). (End)

A264773 Triangle T(n,k) = binomial(4n - 3k, 3n - 2k), 0 <= k <= n.

Original entry on oeis.org

1, 4, 1, 28, 5, 1, 220, 36, 6, 1, 1820, 286, 45, 7, 1, 15504, 2380, 364, 55, 8, 1, 134596, 20349, 3060, 455, 66, 9, 1, 1184040, 177100, 26334, 3876, 560, 78, 10, 1, 10518300, 1560780, 230230, 33649, 4845, 680, 91, 11, 1, 94143280, 13884156, 2035800, 296010, 42504, 5985, 816, 105, 12, 1

Offset: 0

Peter Bala, Nov 30 2015

Riordan array (f(x),x*g(x)), where g(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + ... is the o.g.f. for A002293 and f(x) = g(x)/(4 - 3*g(x)) = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + ... is the o.g.f. for A005810.

More generally, if (R(n,k))n,k>=0 is a proper Riordan array and m is a nonnegative integer and a > b are integers then the array with (n,k)-th element R((m + 1)*n - a*k, m*n - b*k) is also a Riordan array (not necessarily proper). Here we take R as Pascal's triangle and m = a = 3 and b = 2. See A092392, A264772, A264774 and A113139 for further examples.

			Triangle begins
  n\k |       0      1     2    3   4   5   6   7
------+-----------------------------------------------
   0  |       1
   1  |       4      1
   2  |      28      5     1
   3  |     220     36     6    1
   4  |    1820    286    45    7   1
   5  |   15504   2380   364   55   8   1
   6  |  134596  20349  3060  455  66   9   1
   7  | 1184040 177100 26334 3876 560  78  10   1
...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
Peter Bala, A 4-parameter family of embedded Riordan arrays
E. Lebensztayn, On the asymptotic enumeration of accessible automata, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No. 3, 2010, 75-80, Section 2.
R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics CreateSpace Independent Publishing Platform 2006, Section 5.6, ISBN-13: 978-1502925244.

A005810 (column 0), A052203 (column 1), A257633 (column 2), A224274 (column 3), A004331 (column 4). Cf. A002293, A007318, A092392 (C(2n-k,n)), A119301 (C(3n-k,n-k)), A264772, A264774.

/* As triangle: */ [[Binomial(4*n-3*k, 3*n-2*k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 02 2015

A264773:= proc(n,k) binomial(4*n - 3*k, 3*n - 2*k); end proc:
seq(seq(A264773(n,k), k = 0..n), n = 0..10);

A264773[n_,k_] := Binomial[4*n - 3*k, n - k];
Table[A264773[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 06 2024 *)

T(n,k) = binomial(4*n - 3*k, n - k).

O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(4*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(3*n + 1)*binomial(4*n,n)*x^n.

A268315 Decimal expansion of 256/27.

Original entry on oeis.org

9, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8

Offset: 1

Gheorghe Coserea, Feb 01 2016

			9.481481481481481481481481481481...

Exponential growth rate for A000260, A002293, A005810, A006633, A006634, A052203, A066381, A069271, A078516, A078995, A153396, A196678, A268316.

[9] cat &cat[[4, 8, 1]^^45]; // Vincenzo Librandi, Feb 04 2016

Join[{9}, PadRight[{}, 120, {4, 8, 1}]] (* Vincenzo Librandi, Feb 04 2016 *)

PARI
```
1.0 * 256/27
```

More digits from Jon E. Schoenfield, Mar 15 2018

A277170 Numerator of 3F2([3n, -n, n+1],[2n+1, n+1/2], 1).

Original entry on oeis.org

1, -1, 1, -1, 1, -3, 1, -1, 25, -1, 1, -49, 1, -1, 9, -3, 1, -363, 3025, -1, 169, -169, 1, -3, 1, -49, 289, -289, 7225, -361, 361, -361, 1, -1, 1, -529, 529, -529, 330625, -148225, 3025, -675, 9, -3, 7569, -2523, 142129, -409757907, 808201, -961, 8649, -2883, 1, -147

Offset: 0

Seiichi Manyama, Oct 19 2016

Neil Calkin found the closed forms of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) in 2007.

Jonathan Borwein, David Bailey, Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A005810, A066802, A052203, A187364, A277520.

a[n_] := HypergeometricPFQ[{3n, -n, n+1}, {2n+1, n+1/2}, 1] // Numerator; Table[a[n], {n, 0, 53}] (* Jean-François Alcover, Oct 22 2016 *)

(s(n) =) 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) = a(n) / A277520(n).
s(2k) = (A005810(k) / A066802(k))^2 = (((4k)! * (3k)!) / ((6k)! * k!))^2.
s(2k+1) = -1/3 * (A052203(k) / A187364(k))^2 = -1/3 * (((4k+1)! * (3k)!) / ((6k+1)! * k!))^2.

A277520 Denominator of 3F2([3n, -n, n+1],[2n+1, n+1/2], 1).

Original entry on oeis.org

1, 3, 25, 147, 1089, 20449, 48841, 312987, 55190041, 14322675, 100100025, 32065374675, 4546130625, 29873533563, 1859904071089, 4089135109921, 9399479144449, 22568149425822049, 1293753708921104809, 2835106739783283, 3289668853728536041

Offset: 0

Seiichi Manyama, Oct 19 2016

Neil Calkin found the closed forms of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) in 2007.

Jonathan Borwein, David Bailey, Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A005810, A052203, A066802, A187364, A277170 (numerators).

a[n_] := HypergeometricPFQ[{3n, -n, n+1}, {2n+1, n+1/2}, 1] // Denominator;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 22 2016 *)

(s(n) =) 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) = A277170(n) / a(n).
s(2k) = (A005810(k) / A066802(k))^2 = (((4k)! * (3k)!) / ((6k)! * k!))^2.
s(2k+1) = -1/3 * (A052203(k) / A187364(k))^2 = -1/3 * (((4k+1)! * (3k)!) / ((6k+1)! * k!))^2.

A387091 a(n) = binomial(9*n+1,n).

Original entry on oeis.org

1, 10, 171, 3276, 66045, 1370754, 28989675, 621216192, 13442126049, 293052087900, 6426898010533, 141629804643600, 3133614810784185, 69566517009302868, 1548833316392624625, 34569147570568156800, 773240476721553042345, 17328840976366636057110

Offset: 0

Seiichi Manyama, Aug 16 2025

Paolo Xausa, Table of n, a(n) for n = 0..700

Cf. A001700, A045721, A052203, A079589, A079590.

Cf. A062994, A169958.

A387091[n_] := Binomial[9*n + 1, n]; Array[A387091, 20, 0] (* Paolo Xausa, Aug 20 2025 *)

PARI
```
a(n) = binomial(9*n+1, n);
```

a(n) = Sum_{k=0..n} binomial(9*n-k,n-k).
G.f.: 1/(1 - x*g^7*(9+g)) where g = 1+x*g^9 is the g.f. of A062994.
G.f.: g^2/(9-8*g) where g = 1+x*g^9 is the g.f. of A062994.
G.f.: B(x)^2/(1 + 8*(B(x)-1)/9), where B(x) is the g.f. of A169958.
D-finite with recurrence +128*n*(8*n-5)*(4*n-1)*(8*n+1)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n) -81*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n+1)*(3*n-2)*(9*n-4)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Aug 19 2025
a(n) ~ 3^(18*n+3) / (sqrt(Pi*n) * 2^(24*n+5)). - Vaclav Kotesovec, Aug 20 2025

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A119304 Triangle read by rows: T(n,k) = binomial(4n-k,n-k), 0 <= k <= n.

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A264773 Triangle T(n,k) = binomial(4*n - 3*k, 3*n - 2*k), 0 <= k <= n.

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A268315 Decimal expansion of 256/27.

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A277170 Numerator of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1).

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A277520 Denominator of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1).

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A387091 a(n) = binomial(9*n+1,n).

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Formula

A264773 Triangle T(n,k) = binomial(4n - 3k, 3n - 2k), 0 <= k <= n.

A277170 Numerator of 3F2([3n, -n, n+1],[2n+1, n+1/2], 1).

A277520 Denominator of 3F2([3n, -n, n+1],[2n+1, n+1/2], 1).