A026375 -id:A026375 - cp's OEIS frontend

A026374 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 11, 6, 1, 1, 7, 17, 17, 7, 1, 1, 9, 30, 45, 30, 9, 1, 1, 10, 39, 75, 75, 39, 10, 1, 1, 12, 58, 144, 195, 144, 58, 12, 1, 1, 13, 70, 202, 339, 339, 202, 70, 13, 1, 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1

Offset: 0

Clark Kimberling

T(n,k) is number of lattice paths from (0,0) to (n,n-2k) using steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: T(4,1)=6 because we have the following paths from (0,0) to (4,2): UUUD, UUH, UUDU, UDUU, HUU and DUUU. Row sums yield A026383. Column 1 is A032766, column 2 is A026381, column 3 is A026382. - Emeric Deutsch, Jan 25 2004

			Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  4,   4,   1;
  1,  6,  11,   6,    1;
  1,  7,  17,  17,    7,    1;
  1,  9,  30,  45,   30,    9,    1;
  1, 10,  39,  75,   75,   39,   10,    1;
  1, 12,  58, 144,  195,  144,   58,   12,   1;
  1, 13,  70, 202,  339,  339,  202,   70,  13,   1;
  1, 15,  95, 330,  685,  873,  685,  330,  95,  15,  1;
  1, 16, 110, 425, 1015, 1558, 1558, 1015, 425, 110, 16, 1;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Cf. A026383, A051159,A169623, A007318

Cf. A026375 (central terms).

a026374 n k = a026374_tabl !! n !! k
a026374_row n = a026374_tabl !! n
a026374_tabl = [1] : map fst (map snd $ iterate f (1, ([1, 1], [1]))) where
   f (0, (us, vs)) = (1, (zipWith (+) ([0] ++ us) (us ++ [0]), us))
   f (1, (us, vs)) = (0, (zipWith (+) ([0] ++ vs ++ [0]) $
                             zipWith (+) ([0] ++ us) (us ++ [0]), us))
-- Reinhard Zumkeller, Feb 22 2014

p[x, 1] := 1;
p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + 1)^Floor[n/2]];
a = Table[CoefficientList[p[x, n], x], {n, 1, 12}];
Flatten[a] (* Roger L. Bagula and Gary W. Adamson, Dec 04 2009 *)

T(n, k) = number of integer strings s(0), ..., s(n) such that s(0)=0, s(n) = n-2k, where, for 1 <= i <= n, s(i) is even if i is even and |s(i) - s(i-1)| <= 1.
From Emeric Deutsch, Jan 25 2004: (Start)
T(2n, k) = Sum_{j=ceiling(k/2)..k} 3^(2j-k)*binomial(n, j)*binomial(j, k-j);
T(2n+1, k) = T(2n, k-1) + T(2n, k).
G.f.: (1 + z + t*z)/(1 - (1+3*t+t^2)*z^2) = 1 + (1+t)*z + (1+3*t+t^2)*z^2+ ... .
Generating polynomial for row 2n is (1 + 3*t + t^2)^n;
Generating polynomial for row 2n+1 it is (1+t)*(1 + 3*t + t^2)^n. (End)
From Emeric Deutsch, Jan 30 2004: (Start)
T(2n, k) = Sum_{j=ceiling(k/2)..k} 3^(2j-k)*binomial(n, j)*binomial(j, k-j);
T(2n+1, k) = T(2n, k-1) + T(2n, k). (End)

A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)BesselI(0,2*x).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 2, 3, 0, 1, 3, 6, 7, 6, 1, 4, 11, 20, 19, 0, 1, 5, 18, 45, 70, 51, 20, 1, 6, 27, 88, 195, 252, 141, 0, 1, 7, 38, 155, 454, 873, 924, 393, 70, 1, 8, 51, 252, 931, 2424, 3989, 3432, 1107, 0, 1, 9, 66, 385, 1734, 5775, 13236, 18483, 12870, 3139, 252, 1, 10, 83, 560, 2995, 12276, 36645, 73392, 86515, 48620, 8953, 0

Offset: 0

Ilya Gutkovskiy, Sep 20 2017

A(n,k) is the k-th binomial transform of A126869 evaluated at n.

			E.g.f. of column k: A_k(x) =  1 + k*x/1! + (k^2 + 2)*x^2/2! + (k^3 + 6*k)*x^3/3! + (k^4 + 12*k^2 + 6)*x^4/4! + (k^5 + 20*k^3 + 30*k)*x^5/5! + ...
Square array begins:
  1,   1,    1,    1,     1,     1,  ...
  0,   1,    2,    3,     4,     5,  ...
  2,   3,    6,   11,    18,    27,  ...
  0,   7,   20,   45,    88,   155,  ...
  6,  19,   70,  195,   454,   931,  ...
  0,  51,  252,  873,  2424,  5775,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
N. J. A. Sloane, Transforms

Columns k=0..7 give A126869, A002426, A000984, A026375, A081671, A098409, A098410, A104454.
Rows n=0..2 give A000012, A001477, A059100.
Main diagonal gives A186925.
Cf. A292628, A307847.

Table[Function[k, n! SeriesCoefficient[Exp[k x] BesselI[0, 2 x], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/Sqrt[(1 + 2 x - k x) (1 - 2 x - k x)], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

O.g.f. of column k: 1/sqrt( (1 - (k-2)*x)*(1 - (k+2)*x) ).
E.g.f. of column k: exp(k*x)*BesselI(0,2*x).
From Seiichi Manyama, May 01 2019: (Start)
A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + x^2)^n.
A(n,k) = Sum_{j=0..n} (k-2)^(n-j) * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..n} (k+2)^(n-j) * (-1)^j * binomial(n,j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - (k^2-4) * (n-1) * A(n-2,k). (End)
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-2*j) * binomial(n,2*j) * binomial(2*j,j). - Seiichi Manyama, May 04 2019
T(n,k) =  (1/Pi) * Integral_{x = -1..1} (k - 2 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (k + 2 - 4*x^2)^n/sqrt(1 - x^2) dx. - Peter Bala, Jan 27 2020
A(n,k) = (1/4)^n * Sum_{j=0..n} (k-2)^j * (k+2)^(n-j) * binomial(2*j,j) * binomial(2*(n-j),n-j). - Seiichi Manyama, Aug 18 2025

A081606 Numbers having at least one 1 in their ternary representation.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Mar 23 2003

Complement of A005823.
Integers m such that central Delannoy number A001850(m) == 0 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
Integers m such that A026375(m) == 0 (mod 3). - Fabio Visonà, Aug 03 2023

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p.25.
Mathematics Stack Exchange, Proof that integers m in this sequence are such that A026375(m) == 0 (mod 3).

Cf. A007089, A062756, A081609, A081605, A074940.

Select[Range[100],DigitCount[#,3,1]>0&] (* Harvey P. Dale, Nov 26 2022 *)

from itertools import count, islice
def A081606_gen(): # generator of terms
    a = 0
    for n in count(1):
        b = int(bin(n)[2:],3)<<1
        yield from range(a+1,b)
        a = b
A081606_list = list(islice(A081606_gen(),30)) # Chai Wah Wu, Oct 13 2023

from gmpy2 import digits
def A081606(n):
    def f(x):
        s = digits(x>>1,3)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return n+int(s,2)
        return n-1+(int(s[:i] or '0',2)+1<Chai Wah Wu, Oct 29 2024

More terms from Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

A246467 G.f.: 1 / AGM(1-5x, sqrt((1-x)(1-25*x))).

Original entry on oeis.org

1, 9, 121, 2025, 38025, 762129, 15912121, 341621289, 7484845225, 166549691025, 3751508008161, 85341068948529, 1957289174870121, 45199191579030225, 1049893021288265625, 24510327614556266025, 574726636455361317225, 13528549573868347823025, 319541915502909478890625

Offset: 0

Paul D. Hanna, Sep 06 2014

nonn

In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by
1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*[Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k)]^2,
and consists of integer coefficients when 4|(q-p).
Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

			G.f.: A(x) = 1 + 9*x + 121*x^2 + 2025*x^3 + 38025*x^4 + 762129*x^5 +...
where the square-root of the terms yields A026375:
[1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, ...]
the g.f. of which is 1/sqrt((1-x)*(1-5*x)).

Seiichi Manyama, Table of n, a(n) for n = 0..717

Cf. A026375, A168597, A246876, A246906, A248167.

CoefficientList[Series[1/ArithmeticGeometricMean[1-5x,Sqrt[(1-x)(1-25x)]],{x,0,20}],x] (* Harvey P. Dale, Nov 01 2023 *)

{a(n)=polcoeff( 1 / agm(1-5*x, sqrt((1-x)*(1-25*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=sum(k=0,n,binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))

a(n) = A026375(n)^2 = [Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)]^2.
G.f.:  1 / AGM((1-x)*(1+5*x), (1+x)*(1-5*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) ~ 5^(2*n+1) / (4*Pi*n). - Vaclav Kotesovec, Dec 10 2018

A109187 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 6, 0, 12, 0, 1, 0, 30, 0, 20, 0, 1, 20, 0, 90, 0, 30, 0, 1, 0, 140, 0, 210, 0, 42, 0, 1, 70, 0, 560, 0, 420, 0, 56, 0, 1, 0, 630, 0, 1680, 0, 756, 0, 72, 0, 1, 252, 0, 3150, 0, 4200, 0, 1260, 0, 90, 0, 1, 0, 2772, 0, 11550, 0, 9240, 0, 1980, 0, 110, 0, 1

Offset: 0

Emeric Deutsch, Jun 21 2005

A Grand Motzkin path is a path in the half-plane x >= 0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).
From Peter Bala, Feb 11 2017: (Start)
Consider an infinite 1-dimensional integer lattice with an oriented self-loop at each vertex. Then T(n,k) equals the number of walks of length n from a vertex to itself having k loops. There is a bijection between such walks and Grand Motzkin paths which takes a right step and a left step on the lattice to an up step U and a down step D of a Grand Motzkin path respectively, and takes traversing a loop on the lattice to the horizontal step H. See A282252 for the corresponding triangle of walks on a 2-dimensional lattice with self-loops. (End)

			T(3,1)=6 because we have hud,hdu,udh,duh,uhd,dhu, where u=(1,1),d=(1,-1), h=(1,0).
Triangle begins:
n\k   [0]  [1]   [2]   [3]   [4]   [5]   [6]  [7]  [8]  [9] [10]
[0]    1;
[1]    0,   1;
[2]    2,   0,    1;
[3]    0,   6,    0,    1;
[4]    6,   0,   12,    0,    1;
[5]    0,  30,    0,   20,    0,    1;
[6]   20,   0,   90,    0,   30,    0,    1;
[7]    0, 140,    0,  210,    0,   42,    0,   1;
[8]   70,   0,  560,    0,  420,    0,   56,   0,   1;
[9]    0, 630,    0, 1680,    0,  756,    0,  72,   0,   1;
[10] 252,   0, 3150,    0, 4200,    0, 1260,   0,  90,   0,   1;
[11] ...
From _Peter Bala_, Feb 11 2017: (Start)
The infinitesimal generator begins
      0
      0    0
      2    0     0
      0    6     0     0
     -6    0    12     0     0
      0  -30     0    20     0   0
     80    0   -90     0    30   0   0
      0  560     0  -210     0  42   0  0
  -2310    0  2240     0  -420   0  56  0  0
  ....
and equals the generalized exponential Riordan array [log(Bessel_I(0,2x)),x], and so has integer entries. (End)

Gheorghe Coserea, Rows n = 0..100, flattened
Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.

Diagonal of rational function R(x, y, t) = 1/(1 - (x^2 + t*x*y + y^2)) with respect to x,y, i.e., T(n,k) = [(xy)^n*t^k] R(x,y,t). For t=0..7 we have the diagonals: A126869(t=0, column 0), A002426(t=1, row sums), A000984(t=2), A026375(t=3), A081671(t=4), A098409(t=5), A098410(t=6), A104454(t=7).

Cf. A089627, A109188, A105868, A171128, A108666, A282252, A298608.

G:=1/sqrt((1-t*z)^2-4*z^2):Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^k),k=1..n+1) od;
with(PolynomialTools): CL := p -> CoefficientList(simplify(p), x):
C := (n,x) -> binomial(2*n,n)*hypergeom([-n,-n],[-n+1/2],1/2-x/4):
seq(print(CL(C(n,x))), n=0..11); # Peter Luschny, Jan 23 2018

p[0] := 1; p[n_] := GegenbauerC[n, -n , -x/2];
Flatten[Table[CoefficientList[p[n], x], {n, 0, 11}]] (* Peter Luschny, Jan 23 2018 *)

T(n,k) = if ((n-k)%2, 0, binomial(n,k)*binomial(n-k, (n-k)/2));
concat(vector(12, n, vector(n, k, T(n-1, k-1)))) \\ Gheorghe Coserea, Sep 06 2018

G.f.: 1/sqrt((1-tz)^2-4z^2).
Row sums yield the central trinomial coefficients (A002426).
T(2n+1, 0) = 0.
T(2n, 0) = binomial(2n,n) (A000984).
Sum_{k=0..n} k*T(n,k) = A109188(n).
Except for the order, same rows as those of A105868.
Column k has e.g.f. (x^k/k!)*Bessel_I(0,2x). - Paul Barry, Mar 11 2006
T(n,k) = binomial((n+k)/2,k)*binomial(n,(n+k)/2)*(1+(-1)^(n-k))/2. - Paul Barry, Sep 18 2007
Coefficient array of the polynomials P(n,x) = x^n*hypergeom([1/2-n/2,-n/2], [1], 4/x^2). - Paul Barry, Oct 04 2008
G.f.: 1/(1-xy-2x^2/(1-xy-x^2/(1-xy-x^2/(1-xy-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
From Paul Barry, Apr 21 2010: (Start)
Exponential Riordan array [Bessel_I(0,2x), x].
Coefficient array of the polynomials P(n,x) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k, k)*x^(n - 2k).
Diagonal sums are the aerated central Delannoy numbers (A001850 with interpolated zeros). (End)
From Peter Bala, Feb 11 2017: (Start)
T(n,k) = binomial(n,k)*binomial(n-k,floor((n-k)/2))*(1 + (-1)^(n-k))/2.
T(n,k) = (n/k) * T(n-1,k-1).
T(n,k) = the coefficient of H^k in the expansion of (H + U + 1/U)^n.
n-th row polynomial R(n,t) = Sum_{k = 0..floor(n/2)} binomial(n,2*k) * binomial(2*k,k) * t^(n-2*k) = coefficient of x^n in the expansion of (1 + t*x + x^2)^n.
R(n,t) = Sum_{k = 0..n} binomial(n,k)*binomial(2*k,k)*(t - 2)^(n-k).
d/dt(R(n,t)) = n*R(n-1,t).
R(n,t) = (1/Pi) * Integral_{x = 0..Pi} (t + 2*cos(x))^n dx.
Moment representation on a finite interval: R(n,t) = 1/Pi * Integral_{x = t-2 .. t+2} x^n/sqrt((t + 2 - x)*(x - t + 2)) dx.
Recurrence: n*R(n,t) = t*(2*n - 1)*R(n-1,t) - (t^2 - 4)*(n - 1)*R(n-2,t) with R(0,t) = 1 and R(1,t) = t.
R(n,t) = A002426 (t = 1), A000984 (t = 2), A026375 (t = 3), A081671 (t = 4), A098409 (t = 5), A098410 (t = 6) and A104454(t = 7).
The zeros of the row polynomials appear to lie on the imaginary axis in the complex plane. Also, the zeros of R(n,t) and R(n+1,t) appear to interlace on the imaginary axis.
The polynomials R(n,1 + t) are the row polynomials of A171128. (End)
From Peter Luschny, Jan 23 2018: (Start)
These are the coefficients of the polynomials G(n, -n , -x/2) where G(n, a, x) denotes the n-th Gegenbauer polynomial.
These polynomials can also be expressed as C(n, x) = binomial(2*n,n)*hypergeom([-n, -n], [-n+1/2], 1/2-x/4). (End)

A154626 a(n) = 2^n*A001519(n).

Original entry on oeis.org

1, 2, 8, 40, 208, 1088, 5696, 29824, 156160, 817664, 4281344, 22417408, 117379072, 614604800, 3218112512, 16850255872, 88229085184, 461973487616, 2418924584960, 12665653559296, 66318223015936, 347246723858432, 1818207451086848, 9520257811087360

Offset: 0

Paul Barry, Jan 13 2009

Hankel transform of 1,1,3,11,45,... (see A026375). Binomial transform of A015448.
From Gary W. Adamson, Jul 22 2016: (Start)
A production matrix for the sequence is M =
  1, 1, 0, 0, 0, ...
  1, 0, 5, 0, 0, ...
  1, 0, 0, 5, 0, ...
  1, 0, 0, 0, 5, ...
  ...
Take powers of M, extracting the upper left terms; getting
the sequence starting (1, 1, 2, 8, 40, 208, ...). (End)
The sequence is N=5 in an infinite set of INVERT transforms of powers of N prefaced with a "1". (1, 2, 8, 40, ...) is the INVERT transform of (1, 1, 5, 25, 125, ...). The first six of such sequences are shown in A006012 (N=3). - Gary W. Adamson, Jul 24 2016
From Gary W. Adamson, Jul 27 2016: (Start)
The sequence is the first in an infinite set in which we perform the operation for matrix M (Cf. Jul 22 2016), but change the left border successively from (1, 1, 1, 1, ...) then to (1, 2, 2, 2, ...), then (1, 3, 3, 3, ...) ...; generally (1, N, N, N, ...). Extracting the upper left terms of each matrix operation, we obtain the infinite set beginning:
  N=1  (A154626): 1,  2,  8,  40,  208,  1088, ...
  N=2  (A084120): 1,  3, 15,  81,  441,  1403, ...
  N=3  (A180034): 1,  4, 22, 124,  700,  3952, ...
  N=4  (A001653): 1,  5, 29, 169,  985,  5741, ...
  N=5  (A000400): 1,  6, 36, 216, 1296,  7776, ...
  N=6  (A015451): 1,  7, 43, 265, 1633, 10063, ...
  N=7  (A180029): 1,  8, 50, 316, 1996, 12608, ...
  N=8  (A180028): 1,  9, 57, 369, 1285, 15417, ...
  N=9  (.......): 1, 10, 64, 424, 2800, 18496, ...
  N=10 (A123361): 1, 11, 71, 481, 3241, 21851, ...
  N=11 (.......): 1, 12, 78, 540, 3708, 25488, ...
... Each of the sequences begins (1, (N+1), (7*N + 1),
  (40*N + (N-1)^2), ... (End)
The set of infinite sequences shown (Cf. comment of Jul 27 2016), can be generated from the matrices P = [(1,N; 1,5]^n, (N=1,2,3,...) by extracting the upper left terms. Example: N=6 sequence (A015451): (1, 7, 43, 265, ...) can be generated from the matrix P = [(1,6); (1,5)]^n. - Gary W. Adamson, Jul 28 2016

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A006012, A084120, A180034, A001653, A000400, A015451, A180029, A180028, A123362.

[n le 2 select (n) else 6*Self(n-1)-4*Self(n-2): n in [1..25]]; // Vincenzo Librandi, May 15 2015

LinearRecurrence[{6, -4}, {1, 2}, 30] (* Vincenzo Librandi, May 15 2015 *)

Vec((1-4*x) / (1-6*x+4*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017

G.f.: (1 - 4*x) / (1 - 6*x + 4*x^2).
a(n) = A084326(n+1) - 4*A084326(n). - R. J. Mathar, Jul 19 2012
From Colin Barker, Sep 22 2017: (Start)
a(n) = (((3-sqrt(5))^n*(1+sqrt(5)) + (-1+sqrt(5))*(3+sqrt(5))^n)) / (2*sqrt(5)).
a(n) = 6*a(n-1) - 4*a(n-2) for n>1. (End)
E.g.f.: exp(3*x)*(5*cosh(sqrt(5)*x) - sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, Aug 26 2025

A328807 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 3, 1, 3, 8, 1, 3, 9, 20, 1, 3, 11, 27, 48, 1, 3, 15, 45, 81, 112, 1, 3, 23, 93, 195, 243, 256, 1, 3, 39, 225, 639, 873, 729, 576, 1, 3, 71, 597, 2583, 4653, 3989, 2187, 1280, 1, 3, 135, 1665, 11991, 32133, 35169, 18483, 6561, 2816

Offset: 0

Seiichi Manyama, Oct 28 2019

T(n,k) is the constant term in the expansion of (1 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.

For fixed k > 0 is T(n,k) ~ (2^k + 1)^(n + (k-1)/2) / (2^((k-1)^2/2) * sqrt(k) * (Pi*n)^((k-1)/2)). - Vaclav Kotesovec, Oct 28 2019

			Square array begins:
     1,   1,   1,    1,     1,      1, ...
     3,   3,   3,    3,     3,      3, ...
     8,   9,  11,   15,    23,     39, ...
    20,  27,  45,   93,   225,    597, ...
    48,  81, 195,  639,  2583,  11991, ...
   112, 243, 873, 4653, 32133, 260613, ...

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Columns k=0..5 give A001792, A000244, A026375, A002893, A328808, A328809.
Main diagonal gives A328810.
Cf. A309010, A328747, A328748.

T[n_, k_] := Sum[Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

A339710 a(n) = Sum_{k=0..n} binomial(n, k)binomial(2n + k, k)*2^k.

Original entry on oeis.org

1, 7, 81, 1051, 14353, 201807, 2891409, 41976627, 615371169, 9089130967, 135048608401, 2016306678987, 30224723308081, 454603719479839, 6857319231939537, 103694587800440931, 1571449259865571137, 23860205774602899111, 362897293035114695121, 5527773456878667951483

Offset: 0

Yifan Zhang, Dec 13 2020

nonn

Frits Beukers, Some Congruences for Apery Numbers, Mathematisch Instituut, University of Leiden, 1983, pages 1-2.

Seiichi Manyama, Table of n, a(n) for n = 0..838
F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy

Cf. A000079 (Sum(binomial(n, k))), A000984 (Sum(binomial(n, k)^2)), A026375 (Sum(binomial(n, k)*binomial(2*k, k))), A001850 (Sum(binomial(n, k)*binomial(n+k, k))), A005809 (Sum(binomial(n, k)*binomial(2*n, k))).

Cf. A026000, A114496.

Table[Sum[Binomial[n,k]*Binomial[2n+k,k]*2^k,{k,0,n}],{n,0,20}] (* or *)
Table[Hypergeometric2F1[-n,1+2 n,1,-2],{n,0,20}] (* Stefano Spezia, Dec 17 2020 *)

a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n + k, k)*2^k); \\ Michel Marcus, Feb 12 2021

a(n) = 2F1([-n, 1 + 2*n], [1], -2), where 2F1 is the hypergeometric function. - Stefano Spezia, Dec 17 2020
From Vaclav Kotesovec, May 11 2021: (Start)
Recurrence: 3*n*(2*n - 1)*(26*n - 35)*a(n) = (2444*n^3 - 5734*n^2 + 3830*n - 729)*a(n-1) - (n-1)*(2*n - 3)*(26*n - 9)*a(n-2).
a(n) ~ sqrt(3/8 + 11/(8*sqrt(13))) * ((47 + 13*sqrt(13))/6)^n / sqrt(Pi*n). (End)

More terms from Stefano Spezia, Dec 17 2020

A110165 Riordan array (1/sqrt(1-6x+5x^2),(1-3x-sqrt(1-6x+5x^2))/(2x)).

Original entry on oeis.org

1, 3, 1, 11, 6, 1, 45, 30, 9, 1, 195, 144, 58, 12, 1, 873, 685, 330, 95, 15, 1, 3989, 3258, 1770, 630, 141, 18, 1, 18483, 15533, 9198, 3801, 1071, 196, 21, 1, 86515, 74280, 46928, 21672, 7210, 1680, 260, 24, 1, 408105, 356283, 236736, 119154, 44982, 12510, 2484, 333, 27, 1

Offset: 0

Paul Barry, Jul 14 2005

Columns include A026375, A026376 and A026377. Inverse is A110168. Rows sums are A110166. Diagonal sums are A110167.
From Peter Bala, Jan 09 2022: (Start)
This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - 3*x - sqrt(1 - 6*x + 5*x^2))/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + 3*x + x^2. In general the (n,k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

			Rows begin
    1;
    3,   1;
   11,   6,   1;
   45,  30,   9,   1;
  195, 144,  58,  12,   1;
  873, 685, 330,  95,  15,   1;
Production array begins:
  3, 1;
  2, 3, 1;
  0, 1, 3, 1;
  0, 0, 1, 3, 1;
  0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 1, 3, 1;
  ... - _Philippe Deléham_, Feb 08 2014

P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

seq(seq( coeff((x^2 + 3*x + 1)^n, x, n-k), k = 0..n ), n = 0..10); # Peter Bala, Jan 09 2022

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/Sqrt[1-6#+5#^2]&, (1-3#-Sqrt[1-6#+5#^2])/(2#)&, 10] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Number triangle T(n, k) = Sum_{j = 0..n} C(n, j)C(2j, j+k).

T(n,0) = 3*T(n-1,0) + 2*T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k > 0, T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 24 2014

A098473 Triangle T(n,k) read by rows, T(n, k) = binomial(2k, k)binomial(n, k), 0<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 18, 20, 1, 8, 36, 80, 70, 1, 10, 60, 200, 350, 252, 1, 12, 90, 400, 1050, 1512, 924, 1, 14, 126, 700, 2450, 5292, 6468, 3432, 1, 16, 168, 1120, 4900, 14112, 25872, 27456, 12870, 1, 18, 216, 1680, 8820, 31752, 77616, 123552, 115830

Offset: 0

Paul Barry, Sep 09 2004

This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) of degree n and shift 0 for the central binomial sequence A000984. For a definition of Jensen polynomials see a comment in A094436. - Wolfdieter Lang, Jun 25 2019

			Rows begin
  1;
  1,  2;
  1,  4,  6;
  1,  6, 18,  20;
  1,  8, 36,  80,  70;
  1, 10, 60, 200, 350, 252;

Indranil Ghosh, Rows 0..125, flattened
O. T. Dasbach, A natural series for the natural logarithm, Electronic Journal of Combinatorics, (15) 2008 #N5.

Row sums are A026375.
Antidiagonal sums are A026569.
Principal diagonal is A000984.
Cf. A002426, A081671, A098409, A098410, A094436, A126869.

A098473 := proc(n,k) binomial(2*k,k)*binomial(n,k) ; end proc:

Table[Binomial[2k,k]Binomial[n,k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 15 2020 *)

T(n,k)=binomial(2*k, k)*binomial(n, k);
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); /* as triangle */

T(n, k) = binomial(2*k, k)*binomial(n, k).
Sum_{k=0..n} T(n,k)*x^(n-k) = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Sep 28 2007
From Peter Bala, Jun 06 2011: (Start)
O.g.f.: 1/sqrt(1 - t)*1/sqrt(1 - t*(1 + 4*x)) = 1 + (2*x + 1)*t + (1 + 4*x + 6*x^2)*t^2 + ....
Let R_n(x) denote the row generating polynomials of this triangle, which begin
R_1(x) = 1 + 2*x, R_2(x) = 1 + 4*x + 6*x^2, R_3(x) = 1 + 6*x + 18*x^2 + 20*x^3.
Dasbach gives the following slowly converging series for the logarithm function:
log(x)  = Sum_{n >= 1} 1/n*R_n(-1/x), valid for x >= 4.
The polynomials (1 - x)^n*R_n(x/(1 - x)) appear to be the row polynomials of A135091 (see also A171128). (End)

cp's OEIS Frontend

A026374 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.

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A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*BesselI(0,2*x).

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A081606 Numbers having at least one 1 in their ternary representation.

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A246467 G.f.: 1 / AGM(1-5*x, sqrt((1-x)*(1-25*x))).

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A109187 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps.

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A154626 a(n) = 2^n*A001519(n).

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A328807 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

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A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)BesselI(0,2*x).

A246467 G.f.: 1 / AGM(1-5x, sqrt((1-x)(1-25*x))).

A339710 a(n) = Sum_{k=0..n} binomial(n, k)binomial(2n + k, k)*2^k.

A098473 Triangle T(n,k) read by rows, T(n, k) = binomial(2k, k)binomial(n, k), 0<=k<=n.