A025174 -id:A025174 - cp's OEIS frontend

A260687 Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.

1, 2, 2, 6, 15, 9, 24, 104, 144, 64, 120, 770, 1775, 1750, 625, 720, 6264, 20880, 33480, 25920, 7776, 5040, 56196, 250096, 571095, 708295, 453789, 117649, 40320, 554112, 3127040, 9433088, 16486400, 16744448, 9175040, 2097152, 362880, 5973264, 41229324, 156498804

Offset: 1

Peter Bala, Nov 16 2015

Related to A220883 and A251592.

			Triangle begins
...1
...2      2
...6     15       9
..24    104     144      64
.120    770    1775    1750     625
.720   6264   20880   33480   25920    7776
5040  56196  250096  571095  708295  453789  117649
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998

P. Bala, Fractional iteration of a series inversion operator

A000142 (column 0), A000169 (main diagonal), A006675 (column 1). Cf. A001700, A025174, A056856, A163456, A220883, A224274, A251592.

seq(seq(coeff(mul(n*t + k, k = 2 .. n), t, i), i = 0..n-1), n = 1..10);

E.g.f. (with constant term 1 included): A(x,t) = [ 1/x*Revert( x*(1 - x)^t ) ]^(1/t) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + n,n)*x^n = 1 + x + (2 + 2*t)*x^2/2! + (2 + 3*t)*(3 + 3*t)*x^3/3! + (2 + 4*t)*(3 + 4*t)*(4 + 4*t)*x^4/4! + ..., where Revert denotes the series reversion operator with respect to x.
In the notation of the Bala link, A(x,t) = I^t(1/(1 - x)) where I^t is a fractional inversion operator.
A(x,t) = B_(1+t)(x), where B_t(x) is the e.g.f. for A251592 and is the generalized binomial series of Lambert. See Graham et al., Section 5.4 and Section 7.5.
A(x,t)^m = Sum_{n >= 0} m/(n*t + m)*binomial(n*t + n + m - 1,n)*x^n = 1 + m*x + m*(2*t + m + 1)*x^2/2! + m*(3*t + m + 1)*(3*t + m + 2)*x^3/3! + m*(4*t + m + 1)*(4*t + m + 2)*(4*t + m + 3)*x^4/4! + ....
A(x,t)^t = 1 + t*x + t(1 + 3*t)*x^2/2! + t*(1 + 4*t)*(2 + 4*t)*x^3/3! + t*(1 + 5*t)*(2 + 5*t)*(3 + 5*t)*x^4/4! + ... is the e.g.f for A220883 with an extra constant term 1 and an extra factor of t included.
t*log( A(x,t) ) = t*x + t*(1 + 2*t)*x^2/2! + t*(1 + 3*t)*(2 + 3*t)*x^3/3! + t*(1 + 4*t)*(2 + 4*t)*(3 + 4*t)*x^4/4! + ... is the e.g.f for A056856.
For n = 1,2,3,..., the sequence [x^n] A(x,t)^n = [1, (2*t + 3), (3*t + 4)*(3*t + 5)/2!, (4*t + 5)*(4*t + 6)*(4*t + 7)/3!, ...]. This sequence has the following specializations:
t = 0: [1, 3, 10, 35, 126, ...] = A001700 (with different offset).
t = 1: [1, 5, 28, 165, 1001, ...] = A025174.
t = 2: [1, 7, 55, 455, 3876, ...] = A224274.
t = 3: [1, 9, 91, 969, 10626, ...] = A163456.

A309955 a(n) = [x^n] (1 + p(x))^n, where p(x) is the g.f. of A000040.

Original entry on oeis.org

1, 2, 10, 59, 362, 2287, 14707, 95762, 629386, 4166627, 27743445, 185602188, 1246543559, 8399791922, 56762121398, 384513835219, 2610322687850, 17753944125159, 120954505004605, 825274753259894, 5638438272353597, 38569743775323134, 264127692090124488

Offset: 0

Alois P. Heinz, Aug 24 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1185

Cf. A000040, A008485, A008578, A025174, A292871, A309950, A340990.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, ithprime(n),
      (h-> add(b(j, h)*b(n-j, i-h), j=0..n))(iquo(i, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..31);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 1, Prime[n],
     Function[h, Sum[b[j, h]*b[n-j, i-h], {j, 0, n}]][Quotient[i, 2]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

A360546 Triangle read by rows: T(n, m) = (n+1-m)C(2n+2-m, m)C(3n-3m+2, n-m+1)/(2n-m+2).

Original entry on oeis.org

1, 5, 2, 28, 20, 3, 165, 168, 50, 4, 1001, 1320, 588, 100, 5, 6188, 10010, 5940, 1568, 175, 6, 38760, 74256, 55055, 19800, 3528, 280, 7, 245157, 542640, 482664, 220220, 54450, 7056, 420, 8, 1562275, 3922512, 4069800, 2252432, 715715, 130680, 12936, 600, 9

Offset: 0

Vladimir Kruchinin, Feb 11 2023

			Triangle begins:
     1;
     5,     2;
    28,    20,    3;
   165,   168,   50,    4;
  1001,  1320,  588,  100,   5;
  6188, 10010, 5940, 1568, 175, 6;

Cf. A025174, A134481.

A360546 := proc(n, k) m := n-k+1; (1/3)*binomial(3*m, m)*binomial(m + n, k) end:
seq(print(seq(A360546(n, k), k = 0..n)), n = 0..8); # Peter Luschny, Feb 11 2023

Maxima
```
T(n,m):=if n
				
```

G.f.: -1/(2*x) + (sqrt(3)*cot((1/3)*arcsin((3*sqrt(3)*sqrt(x))/(2- 2*x*y))))/ (2*sqrt(x*(-27*x + 4*(-1+x*y)^2))).

A259613 a(n) = binomial(6n,2n)/3, n>0, a(0)=1.

Original entry on oeis.org

1, 5, 165, 6188, 245157, 10015005, 417225900, 17620076360, 751616304549, 32308782859535, 1397281501935165, 60727722660586800, 2650087220696342700, 116043807643289338428, 5096278545356362962504, 224377658168860057076688

Offset: 0

Vladimir Kruchinin, Jun 30 2015

G. C. Greubel, Table of n, a(n) for n = 0..600
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. A001448, A001450, A182959.

[1] cat [Binomial(6*n,2*n)/3: n in [1..20]]; // Vincenzo Librandi, Jul 01 2015

Join[{1}, Table[Binomial[6 n, 2 n]/3, {n, 30}]] (* Vincenzo Librandi, Jul 01 2015 *)

vector(20,n, n--; if (n==0, 1, binomial(6*n,2*n)/3)) \\ Michel Marcus, Jul 01 2015

G.f.: A(x) = 1 + (x*B(x)')/(B(x)) where  B(x) = 2 * (1 + x*B(x)^2)^2 / (1 - 2*x*B(x)^2 + sqrt(1-8*x*B(x)^2)).
a(n) ~ 3^(6*n-1/2) / (sqrt(Pi*n) * 2^(4*n+3/2)). - Vaclav Kotesovec, Jul 01 2015
a(n) = A025174(2*n), n>0. - R. J. Mathar, Jun 07 2016
From Peter Bala, Jun 08 2024: (Start)
a(n) = (9/2)*(6*n-1)*(6*n-5)*(3*n-1)*(3*n-2)/((4*n-1)*(4*n-3)*(2*n-1)*n) * a(n-1) with a(0) = 1 and a(1) = 5.
Right-hand side of the identity (1/3)*Sum_{k = 0..2*n} (-1)^k*binomial(-n, k)* binomial(5*n-k, 2*n-k) = (1/3)*binomial(6*n, 2*n). Compare with the identity Sum_{k = 0..n} (-1)^k*binomial(n, k)*binomial(5*n-k, 2*n-k) = binomial(4*n, 2*n). (End)
From Karol A. Penson, Jan 26 2025: (Start)
G.f. for 3*a(n),a(0)=1, denoted A, expressible entirely by radicals: A = A1 + A2 with
A1 = ((4*sqrt(4 - 27*sqrt(z)) + 12*i*sqrt(3)*z^(1/4))^(1/3) + (4*sqrt(4 - 27*sqrt(z)) - 12*i*sqrt(3)*z^(1/4))^(1/3))/(4*sqrt(4 - 27*sqrt(z))), and
A2 = (1/(4*sqrt(4 + 27*sqrt(z)) + 12*sqrt(3)*z^(1/4))^(1/3) + 1/(4*sqrt(4 + 27*sqrt(z)) - 12*sqrt(3)*z^(1/4))^(1/3))/sqrt(4 + 27*sqrt(z)),
where i = sqrt(-1), the imaginary unit. (End)

A309682 G.f.: C(x)C(2x^2)C(3x^3)*..., where C(x) is the g.f. for A000108.

Original entry on oeis.org

1, 1, 4, 10, 33, 81, 282, 762, 2599, 7979, 27343, 89371, 315256, 1078498, 3857048, 13651786, 49475282, 178736186, 655247192, 2401663838, 8883371016, 32906649488, 122619768860, 457836275272, 1716620421629, 6449729802639, 24308647131627, 91800114425437

Offset: 0

Vaclav Kotesovec, Aug 12 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1669

Cf. A025174, A110143, A322204, A326985.

C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1,
      C(n), add(C(j)*i^j*b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2019

nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*j^k*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*k*x^k])/(2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n / n^(3/2), where c = 1/(2*sqrt(Pi)) * Product_{k>=1} (2^k*(2^(k-1) - sqrt(4^(k-1) - k))/k) = 0.711438694828613555153724789...

A144484 Triangle read by rows: T(n, k) = binomial(3*n+1-k, n-k) for n, k >= 0.

Original entry on oeis.org

1, 4, 1, 21, 6, 1, 120, 36, 8, 1, 715, 220, 55, 10, 1, 4368, 1365, 364, 78, 12, 1, 27132, 8568, 2380, 560, 105, 14, 1, 170544, 54264, 15504, 3876, 816, 136, 16, 1, 1081575, 346104, 100947, 26334, 5985, 1140, 171, 18, 1, 6906900, 2220075, 657800, 177100

Offset: 0

Roger L. Bagula, Oct 12 2008

Previous name: A triangle sequence from a polynomial: p(x,n)=Sum[Binomial[3*n + 1 - m, n - m]*x^m, {m, 0, n}]; p(x,n)=Gamma[2*n+3]*Hypergeometric2F1[1,-n-1-3*n,x]/(Gamma[1+n]*Gamma[2+2*n}).

			{1},
{4, 1},
{21, 6, 1},
{120, 36, 8, 1},
{715, 220, 55, 10, 1},
{4368, 1365, 364, 78, 12, 1},
{27132, 8568, 2380, 560, 105, 14, 1},
{170544, 54264, 15504, 3876, 816, 136, 16, 1},
{1081575, 346104, 100947, 26334, 5985, 1140, 171, 18, 1},
{6906900, 2220075, 657800, 177100, 42504, 8855, 1540, 210, 20, 1},
{44352165, 14307150, 4292145, 1184040, 296010, 65780, 12650, 2024, 253, 22, 1}

M. Jones, Further remarks on the enumeration of graphs, preprint, 2001.

Omer Egecioglu, Timothy Redmond, Charles Ryavec, Almost Product Evaluation of Hankel Determinants, arXiv:0704.3398 [math.CO], 2007.

Cf. A025174 (row sums).

p[x_, n_] = Sum[Binomial[3*n + 1 - m, n - m]*x^m, {m, 0, n}]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]

T(n, k) = binomial(3*n+1-k, n-k);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 13 2018

p(x,n)=Sum[Binomial[3*n + 1 - m, n - m]*x^m, {m, 0, n}];

p(x,n)=Gamma[2*n+3]*Hypergeometric2F1[1,-n-1-3*n,x]/(Gamma[1+n]*Gamma[2+2*n});

New name from Michel Marcus, May 13 2018

A213410 G.f.: exp( Sum_{n>=1} binomial(3n,n)^n/3^n x^n/n ).

Original entry on oeis.org

1, 1, 13, 7330, 185307558, 201002187396362, 9357300769149011773697, 18775362849239140086719414696830, 1631039199744298058694966065590003308698494, 6159916689356522044764167426829149420348399496664634288

Offset: 0

Paul D. Hanna, Jun 10 2012

nonn

Compare to the g.f. G(x) = 1 + x*G(x)^3 of A001764: G(x) = exp( Sum_{n>=1} binomial(3*n,n)/3 * x^n/n ).

			G.f.: A(x) = 1 + x + 13*x^2 + 7330*x^3 + 185307558*x^4 + 201002187396362*x^5 +...
where
log(A(x)) = x + 5^2*x^2/2 + 28^3*x^3/3 + 165^4*x^4/4 + 1001^5*x^5/5 + 6188^6*x^6/6 + 38760^7*x^7/7 +...+ A025174(n)^n*x^n/n +...

Cf A213409, A001764, A025174, A200002.

nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[3*k,k]^k/3^k * b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)

{a(n)=polcoeff(exp(sum(m=1, n, binomial(3*m, m)^m/3^m*x^m/m)+x*O(x^n)), n)}
for(n=0,15,print1(a(n),", "))

a(n) = (1/n) * Sum_{k=1..n} binomial(3*k,k)^k/3^k * a(n-k) for n>0 with a(0)=1.

A277584 a(n) = binomial(3n-1, n-1)^2.

Original entry on oeis.org

0, 1, 25, 784, 27225, 1002001, 38291344, 1502337600, 60101954649, 2440703175625, 100300325150025, 4161829109817600, 174077451630810000, 7330421677037621904, 310467090932230849600, 13214837914326197526784, 564927069263895118093401

Offset: 0

Seiichi Manyama, Oct 22 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..605

Cf. A025174, A060150, A188662.

[Binomial(3*n-1, n-1)^2: n in [0..20]]; // Vincenzo Librandi, Oct 23 2016

Table[Boole[n > 0] Binomial[3 n - 1, n - 1]^2, {n, 0, 16}] (* Michael De Vlieger, Oct 26 2016 *)

a(n) = binomial(3*n-1, n-1)^2; \\ Michel Marcus, Oct 22 2016

a(n) = A025174(n)^2.
a(n) = A188662(n)/9 for n > 0.
Let the number of multisets of length k on n symbols be denoted by ((n, k)) = binomial(n+k-1, k).
a(n) = (Sum_{k=0..n} binomial(n, k)^2 * ((2*n, 2*n - k)))/5 for n > 0.

A333472 a(n) = [x^n] ( c (x/(1 + x)) )^n, where c(x) = (1 - sqrt(1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.

Original entry on oeis.org

1, 1, 3, 13, 59, 276, 1317, 6371, 31131, 153292, 759428, 3780888, 18900389, 94805959, 476945913, 2405454213, 12158471195, 61574325840, 312365992620, 1587052145492, 8074474510884, 41131551386120, 209760563456920, 1070822078321520, 5471643738383781, 27982867986637151

Offset: 0

Peter Bala, Mar 23 2020

Let F(x) = 1 + f(1)*x + f(2)*x^2 + ... be a power series with integer coefficients. The associated sequence u(n) := [x^n] F(x)^n is known to satisfy the Gauss congruences: u(n*p^k) == u(n*p^(k-1)) ( mod p^k ) for any prime p and positive integers n and k. For certain power series F(x), stronger congruences may hold. Examples include F(x) = (1 + x)^2, F(x) = 1/(1 - x) and F(x) = c(x), where c(x) is the o.g.f. of the Catalan numbers A000108. The associated sequences (with some differences of offset) are A000984, A001700 and A025174, respectively.
Here we take F(x) = c(x/(1 + x)) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 21*x^6 + ... (cf. A001006 and A086246) and conjecture that the associated sequence a(n) = [x^n] ( c(x/(1 + x)) )^n satisfies the supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for prime p >= 5 and positive integers n and k. Cf. A333473.
More generally, we conjecture that for any positive integer r and any integer s the sequence a(r,s;n) := [x^(r*n)] ( c(x/(1 + x)) )^(s*n) also satisfies the above congruences.
Note that the sequence b(n) := [x^n] c(x)^n = A025174(n) satisfies the stronger congruences b(n*p^k) == b(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. The sequence d(n) := [x^n] ( (1 + x)*c(x/(1 + x)) )^n = A333093(n) appears to satisfy the same congruences.

			Examples of congruences:
a(11) - a(1) = 3780888 - 1 = (11^2)*31247 == 0 ( mod 11^2 ).
a(3*7) - a(3) = 41131551386120 - 13 = (7^2)*13*23671*2727841 == 0 ( mod 7^2 ).
a(5^2) - a(5) = 27982867986637151 - 276 = (5^4)*13*74687*46113049 == 0 ( mod 5^4 ).

Cf. A000108, A001006, A086246, A106228, A219537, A333093, A333473.

Cat := x -> (1/2)*(1-sqrt(1-4*x))/x:
G := x -> Cat(x/(1+x)):
H := (x,n) -> series(G(x)^n, x, 51):
seq(coeff(H(x, n), x, n), n = 0..25);

Table[SeriesCoefficient[((1 + x - Sqrt[1 - 2*x - 3*x^2]) / (2*x))^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 29 2020 *)

a(n) = [x^n] ( (1 + x - sqrt(1 - 2*x - 3*x^2)) / (2*x) )^n.

a(n) ~ sqrt(((9386 + 1026*sqrt(57))^(1/3) + (9386 - 1026*sqrt(57))^(1/3) - 19)/228) * (((1261 + 57*sqrt(57))^(1/3) + (1261 - 57*sqrt(57))^(1/3) + 10)/6)^n / sqrt(Pi*n). - Vaclav Kotesovec, Mar 29 2020

A344503 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n, k)^2hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).

Original entry on oeis.org

1, 0, -1, 3, 0, -5, 15, 0, -28, 84, 0, -165, 495, 0, -1001, 3003, 0, -6188, 18564, 0, -38760, 116280, 0, -245157, 735471, 0, -1562275, 4686825, 0, -10015005, 30045015, 0, -64512240, 193536720, 0, -417225900, 1251677700, 0, -2707475148, 8122425444, 0, -17620076360

Offset: 0

Peter Luschny, May 23 2021

sign

Inverse binomial convolution of the Motzkin numbers.

Cf. A064189 (Motzkin numbers), A005809, A025174, A344502.

a := n -> add((-1)^(n - k)*binomial(n, k)^2*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4), k = 0..n): seq(simplify(a(n)), n = 0..41);

a(3*n) = binomial(3*n, n) (A005809).
a(3*n - 1) = -binomial(3*n - 1, n - 1) (A025174).
a(3*n - 2) = 0.
Conjecture D-finite with recurrence -18*(2*n+1) *(2*n-1) *(n+1) *a(n) +2*(-36*n^3+554*n^2-1128*n+27) *a(n-1) +6*(-12*n^3-188*n^2+1235*n-1618) *a(n-2) +9*(54*n^3-27*n^2-183*n+320) *a(n-3) +54*(n-3) *(9*n^2-125*n+75) *a(n-4) +81 *(n-3) *(n-4) *(6*n+127) *a(n-5)=0. - R. J. Mathar, Nov 02 2021

cp's OEIS Frontend

A260687 Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Formula

A309955 a(n) = [x^n] (1 + p(x))^n, where p(x) is the g.f. of A000040.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A360546 Triangle read by rows: T(n, m) = (n+1-m)*C(2*n+2-m, m)*C(3*n-3*m+2, n-m+1)/(2*n-m+2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Maxima

Formula

A259613 a(n) = binomial(6*n,2*n)/3, n>0, a(0)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A309682 G.f.: C(x)*C(2*x^2)*C(3*x^3)*..., where C(x) is the g.f. for A000108.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A144484 Triangle read by rows: T(n, k) = binomial(3*n+1-k, n-k) for n, k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A213410 G.f.: exp( Sum_{n>=1} binomial(3*n,n)^n/3^n * x^n/n ).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A277584 a(n) = binomial(3n-1, n-1)^2.

Views

A360546 Triangle read by rows: T(n, m) = (n+1-m)C(2n+2-m, m)C(3n-3m+2, n-m+1)/(2n-m+2).

A259613 a(n) = binomial(6n,2n)/3, n>0, a(0)=1.

A309682 G.f.: C(x)C(2x^2)C(3x^3)*..., where C(x) is the g.f. for A000108.

A213410 G.f.: exp( Sum_{n>=1} binomial(3n,n)^n/3^n x^n/n ).

A333472 a(n) = [x^n] ( c (x/(1 + x)) )^n, where c(x) = (1 - sqrt(1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.

A344503 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n, k)^2hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).