A122898 -id:A122898 - cp's OEIS frontend

A125906 Riordan array (1/(1 + 5x + x^2), x/(1 + 5x + x^2))^(-1); inverse of Riordan array A123967.

1, 5, 1, 26, 10, 1, 140, 77, 15, 1, 777, 540, 153, 20, 1, 4425, 3630, 1325, 254, 25, 1, 25755, 23900, 10509, 2620, 380, 30, 1, 152675, 155764, 79065, 23989, 4550, 531, 35, 1, 919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1

Offset: 0

Philippe Deléham, Feb 04 2007

T(0)=A053121, T(1)=A064189, T(2)=A039598, T(3)=A091965, T(4)=A052179.
Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and five types of steps H=(1,0); example: T(3,1)=77 because we have UDU, UUD, 25 HHU paths, 25 HUH paths and 25 UHH paths. - Philippe Deléham, Sep 25 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
7^n = (n-th row terms) dot (first n+1 terms in 1,2,3,...). Example: 7^3 = 343 = (140, 77, 15, 1) dot (1, 2, 3, 4) = (140 + 154 + 45 + 4) = 343. - Gary W. Adamson, Jun 17 2011
A subset of the "family of triangles" (Deleham comment of Sep 25 2007) is the succession of binomial transforms beginning with triangle A053121, (0,0); giving -> A064189, (1,1); -> A039598, (2,2); -> A091965, (3,3); -> A052179, (4,4); -> A125906, (5,5) ->, etc; generally the binomial transform of the triangle generated from (n,n) = that generated from ((n+1),(n+1)). - Gary W. Adamson, Aug 03 2011
Riordan array (f(x), x*f(x)) where f(x) is the o.g.f. of A182401. - Philippe Deléham, Mar 04 2013

			Triangle begins
       1;
       5,       1;
      26,      10,      1;
     140,      77,     15,      1;
     777,     540,    153,     20,     1;
    4425,    3630,   1325,    254,    25,    1;
   25755,   23900,  10509,   2620,   380,   30,   1;
  152675,  155764,  79065,  23989,  4550,  531,  35,  1;
  919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1;
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins
  5, 1;
  1, 5, 1,;
  0, 1, 5, 1;
  0, 0, 1, 5, 1;
  0, 0, 0, 1, 5, 1;
  0, 0, 0, 0, 1, 5, 1;
  0, 0, 0, 0, 0, 1, 5, 1;
  0, 0, 0, 0, 0, 0, 1, 5, 1;
  0, 0, 0, 0, 0, 0, 0, 1, 5, 1; (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A182401.

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,  T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 5, 5], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Triangle T(5) where T(x) is defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1). Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0). Sum_{k=0..n} T(n,k) = A122898(n).
Sum_{k=0..n} T(n,k)*(k+1) = 7^n. - Philippe Deléham, Mar 26 2007
T(n,0) = A182401(n). - Philippe Deléham, Mar 04 2013
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + 5*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

A091867 Triangle read by rows: T(n,k) = number of Dyck paths of semilength n having k peaks at odd height.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 3, 4, 6, 0, 1, 6, 15, 10, 10, 0, 1, 15, 36, 45, 20, 15, 0, 1, 36, 105, 126, 105, 35, 21, 0, 1, 91, 288, 420, 336, 210, 56, 28, 0, 1, 232, 819, 1296, 1260, 756, 378, 84, 36, 0, 1, 603, 2320, 4095, 4320, 3150, 1512, 630, 120, 45, 0, 1, 1585, 6633, 12760, 15015, 11880, 6930, 2772, 990, 165, 55, 0, 1

Offset: 0

Emeric Deutsch, Mar 10 2004

Number of ordered trees with n edges having k leaves at odd height. Row sums are the Catalan numbers (A000108). T(n,0)=A005043(n). Sum_{k=0..n} k*T(n,k) = binomial(2n-2,n-1).
T(n,k)=number of Dyck paths of semilength n and having k ascents of length 1 (an ascent is a maximal string of consecutive up steps). Example: T(4,2)=6 because we have UdUduud, UduuddUd, uuddUdUd, uudUdUdd, UduudUdd and uudUddUd (the ascents of length 1 are indicated by U instead of u).
T(n,k) is the number of Łukasiewicz paths of length n having k level steps (i.e., (1,0)). A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: T(4,1)=4 because we have HU(2)DD, U(2)HDD, U(2)DHD and U(2)DDH, where H=(1,0), U(1,1), U(2)=(1,2) and D=(1,-1). - Emeric Deutsch, Jan 06 2005
T(n,k) = number of noncrossing partitions of [n] containing k singleton blocks. Also, T(n,k) = number of noncrossing partitions of [n] containing k adjacencies. An adjacency is an occurrence of 2 consecutive integers in the same block (here 1 and n are considered consecutive). In fact, the statistics # singletons and # adjacencies have a symmetric joint distribution.
Exponential Riordan array [e^x*(Bessel_I(0,2x)-Bessel_I(1,2x)),x]. - Paul Barry, Mar 03 2011
T(n,k) is the number of ordered trees having n edges and exactly k nodes with one child. - Geoffrey Critzer, Feb 25 2013
From Tom Copeland, Nov 04 2014: (Start)
Summing the coeff. of the partitions in A134264 for a Lagrange inversion formula (see also A249548) containing (h_1)^k = (1')^k gives this triangle, so this array's o.g.f. H(x,t) = x + t * x^2 + (1 + t^2) * x^3 ... is the inverse of the o.g.f. of A104597 with a sign change, i.e., H^(-1)(x,t) = (x-x^2) / [1 + (t-1)(x-x^2)] = Cinv(x)/[1 + (t-1)Cinv(x)] = P[Cinv(x),t-1] where Cinv(x)= x * (1-x) is the inverse of C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the Catalan numbers A000108, and P(x,t) = x/(1+t*x) with inverse Pinv(x,t) = -P(-x,t) = x/(1-t*x). Therefore,
O.g.f.: H(x,t) = C[Pinv(x,t-1)] = C[P(x,1-t)] = C[x/(1-(t-1)x)] = {1-sqrt[1-4*x/(1-(t-1)x)]}/2 (for A091867). Reprising,
Inverse O.g.f.: H^(-1)(x,t) = x*(1-x) / [1 + (t-1)x(1-x)] = P[Cinv(x),t-1].
From general arguments in A134264, the row polynomials are an Appell sequence with lowering operator d/dt, having the umbral property (p(.,t)+a)^n=p(n,t+a) with e.g.f. = e^(x*t)/w(x), where 1/w(x)= e.g.f. of first column for the Motzkin numbers in A005043. (Mislabeled argument corrected on Jan 31 2016.)
Cf. A124644 (t-shifted polynomials), A026378 (t=-4), A001700 (t=-3), A005773 (t=-2), A126930 (t=-1) and A210736 (t=-1, a(0)=0, unsigned), A005043 (t=0), A000108 (t=1), A007317 (t=2), A064613 (t=3), A104455 (t=4), A030528 (for inverses).
(End)
The sequence of binomial transforms A126930, A005043, A000108, ... in the above comment appears in A126930 and the link therein to a paper by F. Fite et al. on page 42. - Tom Copeland, Jul 23 2016

			T(4,2)=6 because we have (ud)uu(ud)dd, uu(ud)dd(ud), uu(ud)(ud)dd, (ud)(ud)uudd, (ud)uudd(ud) and uudd(ud)(ud) (here u=(1,1), d=(1,-1) and the peaks at odd height are shown between parentheses).
Triangle begins:
   1;
   0,   1;
   1,   0,   1;
   1,   3,   0,   1;
   3,   4,   6,   0,  1;
   6,  15,  10,  10,  0,  1;
  15,  36,  45,  20, 15,  0, 1;
  36, 105, 126, 105, 35, 21, 0, 1;
  ...

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison and Wesley, 1996, page 254 (first edition)

Alois P. Heinz, Rows n = 0..200, flattened
David Callan, On conjugates for set partitions and integer compositions , arXiv:math/0508052 [math.CO], 2005.
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.

Cf. A000108, A001700, A005043, A005773, A007317, A026378, A030528, A064613, A104455, A104597, A124644, A126930, A210736.

T := proc(n,k) if k>n then 0 elif k=n then 1 else (binomial(n+1,k)/(n+1))*sum(binomial(n+1-k,j)*binomial(n-k-j-1,j-1),j=1..floor((n-k)/2)) fi end: seq(seq(T(n,k),k=0..n),n=0..12);
T := (n,k) -> (-1)^(n+k)*binomial(n,k)*hypergeom([-n+k,1/2],[2],4): seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Jul 27 2016
# alternative Maple program:
b:= proc(x, y, t) option remember; expand(`if`(x=0, 1,
      `if`(y>0, b(x-1, y-1, 0)*z^irem(t*y, 2), 0)+
      `if`(y (p-> seq(coeff(p, z, i), i=0..n))(b(2*n, 0$2)):
seq(T(n), n=0..16);  # Alois P. Heinz, May 12 2017

nn=10;cy = ( 1 + x - x y - ( -4x(1+x-x y) + (-1 -x + x y)^2)^(1/2))/(2(1+x-x y)); Drop[CoefficientList[Series[cy,{x,0,nn}],{x,y}],1]//Grid  (* Geoffrey Critzer, Feb 25 2013 *)
Table[Which[k == n, 1, k > n, 0, True, (Binomial[n + 1, k]/(n + 1)) Sum[Binomial[n + 1 - k, j] Binomial[n - k - j - 1, j - 1], {j, Floor[(n - k)/2]}]], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 25 2016 *)

T(n, k) = [binomial(n+1, k)/(n+1)]*Sum_{j=1..floor((n-k)/2)} binomial(n+1-k, j)*binomial(n-k-j-1, j-1) for kn. G.f.=G=G(t, z) satisfies z(1+z-tz)G^2-(1+z-tz)G+1=0. T(n, k)=r(n-k)*binomial(n, k), where r(n)=A005043(n) are the Riordan numbers.
G.f.: 1/(1-xy-x^2/(1-x-xy-x^2/(1-x-xy-x^2/(1-x-xy-x^2/(1-... (continued fraction). - Paul Barry, Aug 03 2009
Sum_{k=0..n} T(n,k)*x^k = A126930(n), A005043(n), A000108(n), A007317(n), A064613(n), A104455(n) for x = -1,0,1,2,3,4 respectively. - Philippe Deléham, Dec 03 2009
Sum_{k=0..n} (-1)^(n-k)*T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Dec 03 2009
E.g.f.: e^(x+xy)*(Bessel_I(0,2x)-Bessel_I(1,2x)). - Paul Barry, Mar 10 2010
From Tom Copeland, Nov 06 2014: (Start)
O.g.f.: H(x,t) = {1-sqrt[1-4x/(1-(t-1)x)]}/2 (shifted index, as given in Copeland's comment, see comp. inverse there).
H(x,t)= x / [1-(C.+(t-1))x] = Sum_{n>=1} (C.+ (t-1))^(n-1)*x^n umbrally, e.g., (a.+b.)^2 = a_0*b_2 + 2 a_1*b1_+ a_0*b_2, where (C.)^n = C_n are the Catalan numbers (1,1,2,5,14,..) of A000108.
This shows directly that the lowering operator for the polynomials is D=d/dt, i.e., D p(n,t)= D(C. + (t-1))^n = n * (C. + (t-1))^(n-1) = n*p(n-1,t), so that the polynomials form an Appell sequence, and that p(n,0) gives a Motzkin sum, or Riordan, number A005043.
(End)
T(n,k) = (-1)^(n+k)*binomial(n,k)*hypergeom([k-n,1/2],[2],4). - Peter Luschny, Jul 27 2016

A360318 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, 12, 74, 466, 2982, 19320, 126390, 833220, 5527190, 36852052, 246751854, 1658106394, 11176100138, 75528743352, 511600414554, 3472363279170, 23609924743590, 160788499672020, 1096566516149790, 7488135911236806, 51193972101241362, 350368409215623192

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360317, A360319, A360321, A360322.

Cf. A085364, A122898.

a(n) = sum(k=0, n, 3^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));

my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)))

G.f.: sqrt( (1-3*x)/(1-7*x) ).
n*a(n) = 2*(5*n-4)*a(n-1) - 21*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/3)^i * a(j) * a(i-j) = (7/3)^n.
a(n) = 2 * A122898(n-1) for n > 0.
a(n) ~ 2 * 7^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A171651 Triangle T, read by rows : T(n,k) = A007318(n,k)*A005773(n+1-k).

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 13, 15, 6, 1, 35, 52, 30, 8, 1, 96, 175, 130, 50, 10, 1, 267, 576, 525, 260, 75, 12, 1, 750, 1869, 2016, 1225, 455, 105, 14, 1, 2123, 6000, 7476, 5376, 2450, 728, 140, 16, 1, 6046, 19107, 27000, 22428, 12096, 4410, 1092, 180, 18, 1

Offset: 0

Philippe Deléham, Dec 14 2009

			Triangle begins:
   1;
   2,   1;
   5,   4,  1;
  13,  15,  6, 1;
  35,  52, 30, 8, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A097692.
Cf. A007318, A005773.
Cf. A168491, A099323, A001405, A005773, A001700, A026378, A005573, A122898.

b:= proc(u, d, t) option remember; `if`(u=0 and d=0, 1/2,
      expand(`if`(u=0, 0, b(u-1, d, 2)*`if`(t=3, x, 1))
      +`if`(d=0, 0, b(u, d-1, `if`(t=2, 3, 1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n+1$2, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Apr 29 2015
# second program:
A171651:= (n, k)-> binomial(n,k)*add((-1)^(n-k-j)*binomial(n-k,j)*binomial(2*j+1,j+1),j=0..n-k): seq(print(seq(A171651(n, k), k=0..n)), n=0..9);  # Mélika Tebni, Dec 16 2023

b[u_, d_, t_] := b[u, d, t] = If[u == 0 && d == 0, 1/2, Expand[If[u == 0, 0, b[u-1, d, 2]*If[t == 3, x, 1]] + If[d == 0, 0, b[u, d-1, If[t == 2, 3, 1]]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n+1, n+1, 1] ];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 21 2016, after Alois P. Heinz *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively.

E.g.f. of column k: exp(x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 16 2023

Corrected by Philippe Deléham, Dec 18 2009

A292630 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(0,2x) + BesselI(1,2x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 10, 13, 6, 1, 5, 17, 35, 35, 10, 1, 6, 26, 75, 126, 96, 20, 1, 7, 37, 139, 339, 462, 267, 35, 1, 8, 50, 233, 758, 1558, 1716, 750, 70, 1, 9, 65, 363, 1491, 4194, 7247, 6435, 2123, 126, 1, 10, 82, 535, 2670, 9660, 23460, 34016, 24310, 6046, 252, 1, 11, 101, 755, 4451, 19846, 63195, 132339, 160795, 92378, 17303, 462

Offset: 0

Ilya Gutkovskiy, Sep 20 2017

A(n,k) is the k-th binomial transform of A001405 evaluated at n.

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 2)*x^2/2! +  (k^3 + 3*k^2 + 6*k + 3)*x^3/3! + (k^4 + 4*k^3 + 12*k^2 + 12*k + 6)*x^4/4! + ...
Square array begins:
   1,   1,    1,     1,     1,     1,  ...
   1,   2,    3,     4,     5,     6,  ...
   2,   5,   10,    17,    26,    37,  ...
   3,  13,   35,    75,   139,   233,  ...
   6,  35,  126,   339,   758,  1491,  ...
  10,  96,  462,  1558,  4194,  9660,  ...

N. J. A. Sloane, Transforms

Columns k=0..5 give A001405, A005773 (with first term deleted), A001700, A026378 (with offset 0), A005573, A122898.

Main diagonal gives A292631.

[seq(seq((k)!*add((m-j)^(j-i)/floor(i/2)!/ceil(i/2)!/(j-i)!,i=0..j),j=0..m), m=0..20)]; # Robert Israel, Sep 20 2017

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

E.g.f. of column k: exp(k*x)*(BesselI(0,2*x) + BesselI(1,2*x)).

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 30, 9, 1, 126, 140, 60, 12, 1, 462, 630, 350, 100, 15, 1, 1716, 2772, 1890, 700, 150, 18, 1, 6435, 12012, 9702, 4410, 1225, 210, 21, 1, 24310, 51480, 48048, 25872, 8820, 1960, 280, 24, 1

Offset: 0

Philippe Deléham, Dec 19 2009

			Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    35,   30,    9,   1;
   126,  140,   60,  12,   1;
   462,  630,  350, 100,  15,  1;
  1716, 2772, 1890, 700, 150, 18, 1;
  ...

Cf. A107230, A171651

Cf. A001405, A001700, A005573, A005773, A026378, A099323, A122898, A168491.

T[n_,k_]:=n!SeriesCoefficient[Exp[2*x]*(BesselI[0,2*x]+BesselI[1,2*x])*x^k / k!,{x,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Dec 23 2023 *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively.
Conjectural g.f.: 1/(2*t)*( sqrt( (1 - x*t)/(1 - (4 + x)*t) ) - 1 ) = 1 + (3 + x)*t + (10 + 6*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013
E.g.f. of column k: exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 23 2023

cp's OEIS Frontend

A125906 Riordan array (1/(1 + 5*x + x^2), x/(1 + 5*x + x^2))^(-1); inverse of Riordan array A123967.

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A091867 Triangle read by rows: T(n,k) = number of Dyck paths of semilength n having k peaks at odd height.

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Mathematica

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A360318 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

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PARI

PARI

Formula

A171651 Triangle T, read by rows : T(n,k) = A007318(n,k)*A005773(n+1-k).

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Programs

Maple

Mathematica

Formula

Extensions

A292630 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) + BesselI(1,2*x)).

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Maple

Mathematica

Formula

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

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Mathematica

Formula

A125906 Riordan array (1/(1 + 5x + x^2), x/(1 + 5x + x^2))^(-1); inverse of Riordan array A123967.

A292630 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(0,2x) + BesselI(1,2x)).