A082758 -id:A082758 - cp's OEIS frontend

A132305 Sum of fifth powers of trinomial coefficients: a(n) = Sum_{k=0..2n} trinomial(n,k)^5 where trinomial(n,k) = [x^k] (1 + x + x^2)^n.

1, 3, 309, 32847, 4775301, 764206503, 131689759209, 23857704965727, 4487745064421061, 869024373004460823, 172218243516497425809, 34778752398142969125627, 7134304921516864247956281, 1482965360342923597534988883, 311760492785929879483633778049

Offset: 0

Paul D. Hanna, Aug 18 2007

Conjecture: the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) holds for all primes p >= 5 and positive integers n and k. - Peter Bala, Aug 29 2025

Cf. A027907, A082758, A132303, A132304.

a(n)=sum(k=0,2*n,polcoeff((1+x+x^2)^n,k)^5)

A273019 a(n) = hypergeom([-2n-1, 1/2], [2], 4) + (2n+1)*hypergeom([-n+1/2, -n], [2], 4).

Original entry on oeis.org

1, 5, 39, 321, 2675, 22483, 190345, 1621413, 13882947, 119385663, 1030434069, 8921880135, 77459553549, 674100041501, 5878674505303, 51361306358401, 449476337521875, 3939287035681807, 34570459724919253, 303749080936528883, 2671775251987354377, 23524418982229636185

Offset: 0

Peter Luschny, May 13 2016

nonn

Bisection of A273020.

Cf. A082758.

a := n -> hypergeom([-2*n-1, 1/2], [2], 4) + (2*n+1)*hypergeom([-n+1/2, -n], [2], 4): seq(simplify(a(n)), n=0..22);

Table[HypergeometricPFQ[{-2*n-1, 1/2}, {2}, 4] + (2*n+1)*HypergeometricPFQ[ {-n+1/2, -n}, {2}, 4], {n, 0, 20}] (* Vaclav Kotesovec, Jul 05 2018 *)

Conjecture: +(241*n+56) *(2*n+1) *(n+1)*a(n) +(482*n^3-25561*n^2+13831*n-56) *a(n-1) +(-48682*n^3+225897*n^2-300131*n+125310) *a(n-2) +9*(n-2) *(2651*n-3183) *(2*n-3) *a(n-3)=0. - R. J. Mathar, May 16 2016
Recurrence (of order 2): (n+1)*(2*n + 1)*(12*n^2 - 19*n + 9)*a(n) = (240*n^4 - 140*n^3 - 42*n^2 - 7*n + 9)*a(n-1) - 9*(n-1)*(2*n - 1)*(12*n^2 + 5*n + 2)*a(n-2). - Vaclav Kotesovec, Jul 05 2018
a(n) ~ 3^(2*n + 3/2) / (2^(3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 05 2018

A273020 a(n) = Sum_{k=0..n} C(n,k)((-1)^n(C(k,n-k)-C(k,n-k-1))+C(n-k,k+1)).

Original entry on oeis.org

1, 1, 3, 5, 19, 39, 141, 321, 1107, 2675, 8953, 22483, 73789, 190345, 616227, 1621413, 5196627, 13882947, 44152809, 119385663, 377379369, 1030434069, 3241135527, 8921880135, 27948336381, 77459553549, 241813226151, 674100041501, 2098240353907, 5878674505303, 18252025766941

Offset: 0

Peter Luschny, May 13 2016

nonn

A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 (2008).

Cf. A005043, A005717, A082758, A273019.

seq(simplify(hypergeom([-n,1/2],[2],4) + n*hypergeom([-n/2+1,-n/2+1/2],[2],4)), n=0..30);

Table[ JacobiP[n, 1, -n-3/2, -7]/(n+1) + GegenbauerC[n-1,-n,-1/2], {n,0,30} ]

def A():
    a, b, c, d, n = 0, 1, 1, -1, 1
    yield 1
    while True:
        yield d + b*(1-(-1)^n)
        n += 1
        a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))
        c, d = d, (3*(n-1)*c-(2*n-1)*d)//n
A273020 = A()
print([next(A273020) for _ in range(31)])

a(n) = JacobiP(n, 1, -n-3/2, -7)/(n+1) + GegenbauerC(n-1, -n, -1/2), with a(0) = 1.
a(n) = hypergeom([-n,1/2], [2], 4) + n*hypergeom([-n/2+1,-n/2+1/2], [2], 4).
a(n) = (-1)^n*A005043(n) + A005717(n).
a(2*n) = A082758(n).
a(2*n+1) = A273019(n).

A280580 Triangle read by rows: T(n,k) = binomial(2n,2k)binomial(2n-2*k,n-k)/(n+1-k) for 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 6, 1, 5, 30, 15, 1, 14, 140, 140, 28, 1, 42, 630, 1050, 420, 45, 1, 132, 2772, 6930, 4620, 990, 66, 1, 429, 12012, 42042, 42042, 15015, 2002, 91, 1, 1430, 51480, 240240, 336336, 180180, 40040, 3640, 120, 1, 4862, 218790, 1312740, 2450448, 1837836, 612612, 92820, 6120, 153, 1

Offset: 0

Werner Schulte, Jan 05 2017

			Triangle begins:
n\k:     0      1       2       3       4      5     6    7  8  . . .
  0:     1
  1:     1      1
  2:     2      6       1
  3:     5     30      15       1
  4:    14    140     140      28       1
  5:    42    630    1050     420      45      1
  6:   132   2772    6930    4620     990     66     1
  7:   429  12012   42042   42042   15015   2002    91    1
  8:  1430  51480  240240  336336  180180  40040  3640  120  1
  etc.
T(3,2) = binomial(6,4) * binomial(2,1) / (3+1-2) = 15 * 2 / 2 = 15. - _Indranil Ghosh_, Feb 15 2017

Indranil Ghosh, Rows 0..100 of triangle, flattened

Row sums are A026945.
Triangle related to A000108, A001006, A001263, and A039599.
Cf. A000007, A001003, A005568, A103365, A238390.

T(n,k) = A001263(n+1,k+1)*A000108(n)/A000108(k) for 0 <= k <= n.
T(n,k) = binomial(2*n,2*k)*A000108(n-k) for 0 <= k <= n.
T(n,k) = A039599(n,k)*binomial(n+1+k,2*k+1)/(n+1-k) for 0 <= k <= n.
Recurrences: T(n,0) = A000108(n) and (1) T(n,k) = T(n,k-1)*(n+1-k)*(n+2-k)/ (2*k*(2*k-1)) for 0 < k <= n, (2) T(n,k) = T(n-1,k-1)*n*(2*n-1)/(k*(2*k-1)).
The row polynomials p(n,x) = Sum_{k=0..n} T(n,k)*x^(2*k) satisfy the recurrence equation p"(n,x) = 2*n*(2*n-1)*p(n-1,x) with initial value p(0,x) = 1 ( n > 0, p" is the second derivative of p ), and Sum_{n>=0} p(n,x)*t^(2*n)/((2*n)!) = cosh(x*t)*(Sum_{n>=0} A000108(n)*t^(2*n)/((2*n)!)).
Conjectures: (1) Sum_{k=0..n} (-1)^k*T(n,k)*A238390(k) = A000007(n);
  (2) Antidiagonal sums equal A001003(n);
  (3) Matrix inverse equals T(n,k)*A103365(n+1-k).
Sum_{k=0..n} (n+1-k)*T(n,k) = A002426(2*n) = A082758(n).
Sum_{k=0..n} T(n,k)*A000108(k) = A000108(n)*A000108(n+1) = A005568(n).
Matrix product: Sum_{i=0..n} T(n,i)*T(i,k) = T(n,k)*A000108(n+1-k) for 0<=k<=n.
T(n,k) = A097610(2*n,2*k) for 0 <= k <= n.
Sum_{k=0..n} (k+1)*T(n,k)*A000108(k) = binomial(2*n+1,n)*A000108(n).

A102319 A mean binomial transform of the central binomial numbers.

Original entry on oeis.org

1, 2, 7, 26, 107, 462, 2065, 9438, 43811, 205622, 972917, 4631838, 22157525, 106406978, 512629551, 2476289106, 11989326771, 58163714118, 282662269717, 1375801775214, 6705710840657, 32724623955882, 159880046446611

Offset: 0

Paul Barry, Jan 04 2005

Second binomial transform of A082758 (with interpolated zeros).

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

A102319 := proc(n)
    add(binomial(n, k)*binomial(2*k, k)*(1+(-1)^(n-k))/2,k=0..n) ;
end proc: # R. J. Mathar, Feb 20 2015

CoefficientList[Series[(1/Sqrt[1-6*x+5*x^2]+1/Sqrt[1-2*x-3*x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 29 2013 *)

x='x+O('x^50); Vec((1/sqrt(1-6*x+5*x^2) + 1/sqrt(1-2*x-3*x^2))/2) \\ G. C. Greubel, Mar 16 2017

G.f.: (1/sqrt(1-6*x+5*x^2) + 1/sqrt(1-2*x-3*x^2))/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*binomial(2*(n-2*k), n-2*k).
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*(1+(-1)^(n-k))/2.
E.g.f.: cosh(x)*exp(2*x)*I_0(2x). - Paul Barry, May 01 2005
a(n) ~ 5^(n+1/2)/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 29 2013
Conjecture: n*(n-1)*a(n) -4*(n-1)*(3*n-4)*a(n-1) +(53*n^2-221*n+232)*a(n-2) +8*(-13*n^2+85*n-134)*a(n-3) +(51*n^2-563*n+1308)*a(n-4) +4*(29*n-93)*(n-4)*a(n-5) -105*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, Feb 20 2015
Conjecture:+n*(n-1)*(12*n^2-48*n+41)*a(n) -8*(n-1)*(12*n^3-54*n^2+65*n-17)*a(n-1) +2*(84*n^4-504*n^3+1025*n^2-775*n+131)*a(n-2) +8*(n-2)*(12*n^3-54*n^2+65*n-20)*a(n-3) -15*(n-2)*(n-3)*(12*n^2-24*n+5)*a(n-4)=0. - R. J. Mathar, Feb 20 2015

A122935 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 19, 10, 1, 0, 1, 15, 45, 45, 15, 1, 0, 1, 21, 90, 141, 90, 21, 1, 0, 1, 28, 161, 357, 357, 161, 28, 1, 0, 1, 36, 266, 784, 1107, 784, 266, 36, 1, 0, 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1, 0, 1, 55, 615, 2850, 6765, 8953

Offset: 0

Philippe Deléham, Oct 30 2006

Subtriangle (1 <= k <= n) is in A056241.

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,   1;
  0, 1,  6,   6,    1;
  0, 1, 10,  19,   10,    1;
  0, 1, 15,  45,   45,   15,    1;
  0, 1, 21,  90,  141,   90,   21,    1;
  0, 1, 28, 161,  357,  357,  161,   28,    1;
  0, 1, 36, 266,  784, 1107,  784,  255,   36,   1;
  0, 1, 45, 414, 1554, 2907, 2907, 1554,  414,  45,  1;
  0, 1, 55, 615, 2850, 6765, 8953, 6765, 2850, 615, 55, 1;

Columns: A000007, A000012, A000217, A005712, A005714, A005716.

Cf. A027907, A056241.

T(2*k-1,k) = A082758(k-1)for k >= 1.
Sum_{k=0..n} T(n,k) = A124302(n); see also A007051.
Sum_{k=0..n} (-1)^(n-k)*T(n,k) = A117569(n).
G.f.: (1-x*(y+2)+x^2)/(1-2x*(1+y)+(1+y+y^2)*x^2). - Philippe Deléham, Oct 30 2011

A291080 Irregular triangle read by rows: T(n,m) = number of lattice paths of type {A^H}_R terminating at point (n, m).

Original entry on oeis.org

1, 3, 2, 1, 19, 16, 10, 4, 1, 141, 126, 90, 50, 21, 6, 1, 1107, 1016, 784, 504, 266, 112, 36, 8, 1, 8953, 8350, 6765, 4740, 2850, 1452, 615, 210, 55, 10, 1, 73789, 69576, 58278, 43252, 28314, 16236, 8074, 3432, 1221, 352, 78, 12, 1, 616227, 585690, 502593, 388752, 270270, 168168, 93093, 45474, 19383, 7098, 2184, 546, 105, 14, 1

Offset: 0

N. J. A. Sloane, Aug 19 2017

			Triangle begins:
  1;
  3,2,1;
  19,16,10,4,1;
  141,126,90,50,21,6,1;
  1107,1016,784,504,266,112,36,8,1;
  8953,8350,6765,4740,2850,1452,615,210,55,10,1;
  73789,69576,58278,43252,28314,16236,8074,3432,1221,352,78,12,1;
  616227,585690,502593,388752,270270,168168,93093,45474,19383,7098,2184,546,105,14,1;
  ...

Rezig Boualam and Moussa Ahmia, Log-concavity and strong q-log-convexity for some generalized triangular arrays, arXiv:2409.18886 [math.CO], 2024. See pp. 3,6.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

First column is A082758.

cp's OEIS Frontend

A132305 Sum of fifth powers of trinomial coefficients: a(n) = Sum_{k=0..2n} trinomial(n,k)^5 where trinomial(n,k) = [x^k] (1 + x + x^2)^n.

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A273019 a(n) = hypergeom([-2*n-1, 1/2], [2], 4) + (2*n+1)*hypergeom([-n+1/2, -n], [2], 4).

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A273020 a(n) = Sum_{k=0..n} C(n,k)*((-1)^n*(C(k,n-k)-C(k,n-k-1))+C(n-k,k+1)).

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A280580 Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.

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A102319 A mean binomial transform of the central binomial numbers.

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A122935 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A291080 Irregular triangle read by rows: T(n,m) = number of lattice paths of type {A^H}_R terminating at point (n, m).

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A273019 a(n) = hypergeom([-2n-1, 1/2], [2], 4) + (2n+1)*hypergeom([-n+1/2, -n], [2], 4).

A273020 a(n) = Sum_{k=0..n} C(n,k)((-1)^n(C(k,n-k)-C(k,n-k-1))+C(n-k,k+1)).

A280580 Triangle read by rows: T(n,k) = binomial(2n,2k)binomial(2n-2*k,n-k)/(n+1-k) for 0<=k<=n.