A100071 -id:A100071 - cp's OEIS frontend

A134757 A007318 * A100071.

1, 3, 11, 37, 123, 401, 1293, 4131, 13107, 41353, 129873, 406319, 1267093, 3940431, 12224579, 37845117, 116944371, 360771417, 1111332129, 3418840431, 10504903809, 32242682787, 98863833159, 302863592073, 927025884477, 2835306153351, 8665554849903

Offset: 1

Gary W. Adamson, Nov 08 2007

nonn

Also A007318^(-1) * A037965. - Gary W. Adamson, Nov 10 2007

			a(3) = 11 = (1, 2, 1) dot (1, 2, 6) = (1 + 4 + 6), where A100071 = (1, 2, 6, 12, 30, ...).
a(3) = 11 = (1, -2, 1) dot (1, 4, 18) = (1 - 8 + 18). - _Gary W. Adamson_, Nov 10 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A037965, A100071, A107965.

A134757:= func< n | (&+[(-1)^(n-k-1)*(k+1)^2*Binomial(n-1,k)*Catalan(k) : k in [0..n-1]]) >;
[A134757(n): n in [1..40]]; // G. C. Greubel, May 28 2024

a[n_]:= a[n]= Sum[(-1)^(n-k-1)*Binomial[n-1,k]*(k+1)*Binomial[2*k, k], {k,0,n-1}];
Table[a[n], {n,40}] (* G. C. Greubel, May 28 2024 *)

def A134757(n): return sum((-1)^(n-k-1)*(k+1)*binomial(n-1,k)*binomial( 2*k, k) for k in range(n))
[A134757(n) for n in range(1,41)] # G. C. Greubel, May 28 2024

Binomial transform of A100071 starting [1, 2, 6, 12, 30, ...].
Inverse binomial transform of A037965 starting [1, 4, 18, 80, 350, ...].
a(n) = (n-1)! * [x^(n-1)] exp(x)*((1 + 2*x)*BesselI(0, 2*x) + 2*x*BesselI(1, 2*x)) for n>0, a(0) = 0. - Peter Luschny, Aug 26 2012
D-finite with recurrence (n-1)*a(n) = 3*(n-1)*a(n-1) +(n+1)*a(n-2) -3*(n-3)*a(n-3). - R. J. Mathar, Nov 09 2021
G.f.: x*(1-x)/((1-3*x)*sqrt((1+x)*(1-3*x))). - G. C. Greubel, May 28 2024

A215619 a(n) is the number of consecutive terms of A100071, beginning with index n, which are divisible by n.

Original entry on oeis.org

4, 1, 6, 1, 8, 1, 4, 1, 12, 5, 14, 1, 4, 1, 18, 1, 20, 1, 4, 1, 24, 1, 6, 1, 4, 1, 30, 21, 32, 1, 12, 1, 8, 1, 38, 1, 14, 1, 42, 1, 44, 1, 6, 1, 48, 1, 8, 1, 4, 1, 54, 1, 6, 9, 4, 1, 60, 1, 62, 1, 4, 1, 6, 1, 68, 1, 4, 1, 72, 1, 74, 1, 4, 1, 12, 1, 80, 1, 4, 1

Offset: 3

Vladimir Shevelev, Aug 17 2012

nonn

a(n) = n+1 iff n is prime.
a(n) = 1  iff n in { A067315 }.
1 <= a(n) <= n+1.
{ n : a(2n)>1 } = { A058008 } \ { 1 }.

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Cf. A000040, A067315, A100071.

b:= proc(n) b(n):= n * binomial(n-1, floor((n-1)/2)) end:
a:= proc(n) local k;
      for k from 0 while irem(b(n+k), n)=0 do od; k
    end:
seq (a(n), n=3..100);  # Alois P. Heinz, Aug 17 2012

b[n_] := n Binomial[n-1, Floor[(n-1)/2]];
a[n_] := Module[{k = 0}, While[Mod[b[n+k], n] == 0, k++]; k];
a /@ Range[3, 100] (* Jean-François Alcover, Nov 22 2020, after Alois P. Heinz *)

A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 40116600, 1163381400, 155117520, 4808643120, 601080390, 19835652870, 2333606220

Offset: 0

Labos Elemer, Jul 25 2000

nonn

a(n) is the number of 'swinging orbitals' which are enumerated by the trinomial n over [floor(n/2), n mod 2, floor(n/2)].
Similar to but different from A001405(n) = binomial(n, floor(n/2)), a(n) = lcm(A001405(n-1), A001405(n)) (for n>0).
A055773(n) divides a(n), A001316(floor(n/2)) divides a(n).
Exactly p consecutive multiples of p follow the least positive multiple of p if p is an odd prime. Compare with the similar property of A100071. - Peter Luschny, Aug 27 2012
a(n) is the number of vertices of the polytope resulting from the intersection of an n-hypercube with the hyperplane perpendicular to and bisecting one of its long diagonals. - Didier Guillet, Jun 11 2018 [Edited by Peter Munn, Dec 06 2022]

			a(10) = 10!/5!^2 = trinomial(10,[5,0,5]);
a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Didier Guillet, On swinging factorials and the lonely runner conjecture (Text in French).
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Orbitals.
Peter Luschny, Swinging Factorial.

Bisections are A000984 and A002457.

Cf. A000142, A001405, A000188, A055772, A056042, A211226, A162246, A189231, A212303.

[(Factorial(n)/(Factorial(Floor(n/2)))^2): n in [0..40]]; // Vincenzo Librandi, Sep 11 2011

SeriesCoeff := proc(s,n) series(s(w,n),w,n+2);
convert(%,polynom); coeff(%,w,n) end;
a1 := proc(n) local k;
2^(n-(n mod 2))*mul(k^((-1)^(k+1)),k=1..n) end:
a2 := proc(n) option remember;
`if`(n=0,1,n^irem(n,2)*(4/n)^irem(n+1,2)*a2(n-1)) end;
a3 := n -> n!/iquo(n,2)!^2;
g4 := z -> BesselI(0,2*z)*(1+z);
a4 := n -> n!*SeriesCoeff(g4,n);
g5 := z -> (1+z/(1-4*z^2))/sqrt(1-4*z^2);
a5 := n -> SeriesCoeff(g5,n);
g6 := (z,n) -> (1+z^2)^n+n*z*(1+z^2)^(n-1);
a6 := n -> SeriesCoeff(g6,n);
a7 := n -> combinat[multinomial](n,floor(n/2),n mod 2,floor(n/2));
h := n -> binomial(n,floor(n/2)); # A001405
a8 := n -> ilcm(h(n-1),h(n));
F := [a1, a2, a3, a4, a5, a6, a7, a8];
for a in F do seq(a(i), i=0..32) od;

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 02 2010 *)
f[n_] := If[OddQ@n, n*Binomial[n - 1, (n - 1)/2], Binomial[n, n/2]]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 10 2010 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; (* or, twice faster: *) sf[n_] := n!/Quotient[n, 2]!^2; Table[sf[n], {n, 0, 32}] (* Jean-François Alcover, Jul 26 2013, updated Feb 11 2015 *)

a(n)=n!/(n\2)!^2 \\ Charles R Greathouse IV, May 02 2011

def A056040():
    r, n = 1, 0
    while True:
        yield r
        n += 1
        r *= 4/n if is_even(n) else n
a = A056040(); [next(a) for i in range(36)]  # Peter Luschny, Oct 24 2013

a(n) = n!/floor(n/2)!^2. [Essentially the original name.]
a(0) = 1, a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1) for n>=1.
E.g.f.: (1+x)*BesselI(0, 2*x). - Vladeta Jovovic, Jan 19 2004
O.g.f.: a(n) = SeriesCoeff_{n}((1+z/(1-4*z^2))/sqrt(1-4*z^2)).
P.g.f.: a(n) = PolyCoeff_{n}((1+z^2)^n+n*z*(1+z^2)^(n-1)).
a(2n+1) = A046212(2n+1) = A100071(2n+1). - M. F. Hasler, Jan 25 2012
a(2*n) = binomial(2*n,n); a(2*n+1) = (2*n+1)*binomial(2*n,n). Central terms of triangle A211226. - Peter Bala, Apr 10 2012
D-finite with recurrence: n*a(n) + (n-2)*a(n-1) + 4*(-2*n+3)*a(n-2) + 4*(-n+1)*a(n-3) + 16*(n-3)*a(n-4) = 0. - Alexander R. Povolotsky, Aug 17 2012
Sum_{n>=0} 1/a(n) = 4/3 + 8*Pi/(9*sqrt(3)). - Alexander R. Povolotsky, Aug 18 2012
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+1)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
Central column of the coefficients of the swinging polynomials A162246. - Peter Luschny, Oct 22 2013
a(n) = Sum_{k=0..n} A189231(n, 2*k). (Cf. A212303 for the odd case.) - Peter Luschny, Oct 30 2013
a(n) = hypergeometric([-n,-n-1,1/2],[-n-2,1],2)*2^(n-1)*(n+2). - Peter Luschny, Sep 22 2014
a(n) = 4^floor(n/2)*hypergeometric([-floor(n/2), (-1)^n/2], [1], 1). - Peter Luschny, May 19 2015
Sum_{n>=0} (-1)^n/a(n) = 4/3 - 4*Pi/(9*sqrt(3)). - Amiram Eldar, Mar 10 2022

Extended and edited by Peter Luschny, Jun 28 2009

A085750 Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

Original entry on oeis.org

0, -1, 4, -12, 32, -80, 192, -448, 1024, -2304, 5120, -11264, 24576, -53248, 114688, -245760, 524288, -1114112, 2359296, -4980736, 10485760, -22020096, 46137344, -96468992, 201326592, -419430400, 872415232, -1811939328, 3758096384, -7784628224, 16106127360

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 21 2003

The determinant of the distance matrix of a tree with vertex set {1,2,...,n}. The distance matrix is the n X n matrix in which the (i,j)-term is the number of edges in the unique path from vertex i to vertex j. [The matrix A in the definition is the distance matrix of the path-tree 1-2-...-n.]
Hankel transform of A100071. Also Hankel transform of C(2n-2,n-1)(-1)^(n-1). Inverse binomial transform of -n. - Paul Barry, Jan 11 2007
Pisano period lengths: 1, 1, 3, 1, 20, 3, 42, 1, 9, 20, 55, 3,156, 42, 60, 1,136, 9,171, 20, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Emmanuel Briand, Luis Esquivias, Álvaro Gutiérrez, Adrián Lillo, and Mercedes Rosas, Determinant of the distance matrix of a tree, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024) Article #29, 12 pp.
R. L. Graham and H. O. Pollak, On the addressing problem for loop switching, Bell System Tech. J., 50, 1971, 2495-2519.
Tanya Khovanova, Recursive Sequences
R. Merris, The distance spectrum of a tree, J. Graph Theory, 14, No. 3, 1990,365-369.
Index entries for linear recurrences with constant coefficients, signature (-4,-4).

Essentially the same as A001787.

Cf. A085807, A100071, A203993, A204249, A278845, A278847.

seq((-1)^(n-1)*(n-1)*2^(n-2), n = 1 .. 31);

Table[-(-1)^n*2^(n - 2)*(n - 1), {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{-4,-4},{0,-1},40] (* Harvey P. Dale, Apr 14 2014 *)
CoefficientList[Series[-x/(1 + 2 x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 15 2014 *)

a(n) = (-1)^n*(1-n)<<(n-2) \\ Charles R Greathouse IV, Sep 30 2022

a(n) = (-1)^(n+1) * (n-1) * 2^(n-2) = (-1)^(n+1) * A001787(n-1).
G.f.: -x/(1+2x)^2. - Paul Barry, Jan 11 2007
a(n) = -4*a(n-1) - 4*a(n-2); a(1) = 0, a(1) = -1. - Philippe Deléham, Nov 03 2008
E.g.f.: -x*exp(-2*x). - Stefano Spezia, Sep 30 2022

More terms from Philippe Deléham, Nov 16 2008

A046212 First numerator and then denominator of central elements of Leibniz's Harmonic Triangle.

Original entry on oeis.org

1, 1, 1, 6, 1, 30, 1, 140, 1, 630, 1, 2772, 1, 12012, 1, 51480, 1, 218790, 1, 923780, 1, 3879876, 1, 16224936, 1, 67603900, 1, 280816200, 1, 1163381400, 1, 4808643120, 1, 19835652870, 1, 81676217700, 1, 335780006100, 1, 1378465288200, 1

Offset: 1

Mohammad K. Azarian

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506.

Cf. A002457.

a(2n+1) = A056040(2n+1) = A100071(2n+1). - M. F. Hasler, Jan 25 2012

More terms from James Sellers, Dec 13 1999

A037965 a(n) = nbinomial(2n-2, n-1).

Original entry on oeis.org

0, 1, 4, 18, 80, 350, 1512, 6468, 27456, 115830, 486200, 2032316, 8465184, 35154028, 145608400, 601749000, 2481880320, 10218366630, 42004911960, 172427570700, 706905276000, 2894777105220, 11841673237680, 48394276165560, 197602337462400, 806190092077500

Offset: 0

N. J. A. Sloane

a(n+1) is the convolution of A000984 and A081294. - Paul Barry, Sep 18 2008

The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972.

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

Cf. A000984, A001622, A081294, A109188 (inverse binomial transform).

Cf. A000108, A100071, A134757.

[0] cat [n^2*Catalan(n-1): n in [1..30]]; // G. C. Greubel, Jun 19 2022

a[n_]:= n*Binomial[2*n-2, n-1]; Array[a, 30, 0] (* Amiram Eldar, Mar 10 2022 *)

a(n) = n*binomial(2*n-2, n-1); \\ Joerg Arndt, Sep 04 2017

[n^2*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 19 2022

Assuming offset -1 here and offset 0 in A134757, A134757 is the inverse binomial transform of this sequence. - Gary W. Adamson, Nov 08 2007
G.f.: Hypergeometric2F1([1/2, 2], [1], 4*x). - Paul Barry, Sep 03 2008
From Paul Barry, Sep 18 2008: (Start)
G.f.: x*(1-2*x)/(1-4*x)^(3/2);
a(n+1) = Sum_{k=0..n} binomial(2*k,k)*(4^(n-k) + 0^(n-k))/2. (End)
D-finite with recurrence (n-1)*a(n) - 2*(3*n-4)*a(n-1) + 4*(2*n-5)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
E.g.f.: x*exp(2*x)*BesselI(0,2*x). - Ilya Gutkovskiy, Aug 22 2018
a(n) = n*A000984(n-1). - R. J. Mathar, Nov 08 2021
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi/(3*sqrt(3)) - Pi^2/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(phi)/sqrt(5) - 4*log(phi)^2, where phi is the golden ratio (A001622). (End)

More terms from Zerinvary Lajos, Oct 02 2007

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

Original entry on oeis.org

1, -2, 6, 3, -24, 30, -4, 60, -180, 140, 5, -120, 630, -1120, 630, -6, 210, -1680, 5040, -6300, 2772, 7, -336, 3780, -16800, 34650, -33264, 12012, -8, 504, -7560, 46200, -138600, 216216, -168168, 51480, 9, -720, 13860, -110880, 450450, -1009008, 1261260, -823680, 218790

Offset: 0

N. J. A. Sloane, Jan 17 2020

Tables I, III, IV on pages 92 and 93 of Ser have integer entries and are A331430, A331431 (the present sequence), and A331432.

Given the system of equations 1 = Sum_{j=0..n} H(i, j) * x(j) for i = 2..n+2 where H(i,j) = 1/(i+j-1) for 1 <= i,j <= n is the n X n Hilbert matrix, then the solutions are x(j) = T(n, j). - Michael Somos, Mar 20 2020 [Corrected by Petros Hadjicostas, Jul 09 2020]

			Triangle begins:
   1;
  -2,    6;
   3,  -24,    30;
  -4,   60,  -180,     140;
   5, -120,   630,   -1120,     630;
  -6,  210, -1680,    5040,   -6300,     2772;
   7, -336,  3780,  -16800,   34650,   -33264,   12012;
  -8,  504, -7560,   46200, -138600,   216216, -168168,   51480;
   9, -720, 13860, -110880,  450450, -1009008, 1261260, -823680, 218790;
  ...

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. See Table III.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Buhl, Book review: J. Ser - Les calculs formels des séries de factorielles, L'Enseignement Mathématique, 32 (1933), p. 275.
L. A. MacColl, Review: J. Ser, Les calculs formels des séries de factorielles, Bull. Amer. Math. Soc., 41(3) (1935), p. 174.
L. M. Milne-Thomson, Review of Les calculs formels des séries de factorielles. By J. Ser. Pp. vii, 98. 20 fr. 1933. (Gauthier-Villars), The Mathematical Gazette, Vol. 18, No. 228 (May, 1934), pp. 136-137.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages.)

Columns 1 is A331433 or equally A007531, column 2 is A331434 or equally A054559; the last three diagonals are A002738, A002736, A002457.

Cf. A000290 (row sums), A002457,, A100071, A108666 (alternating row sums), A109188 (diagonal sums), A331322, A331323, A331430, A331432.

[(-1)^(n+k)*(k+1)*(2*k+1)*Binomial(n+k+1,n-k)*Catalan(k): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 22 2022

gf := k -> (1+x)^(-2*(k+1)): ser := k -> series(gf(k), x, 32):
T := (n, k) -> ((2*k+1)!/(k!)^2)*coeff(ser(k), x, n-k):
seq(seq(T(n,k), k=0..n),n=0..7); # Peter Luschny, Jan 18 2020
S:=(n,k)->(-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!);
rho:=n->[seq(S(n,k),k=0..n)];
for n from 0 to 14 do lprint(rho(n)); od: # N. J. A. Sloane, Jan 18 2020

Table[(-1)^(n+k)*(n+k+1)*Binomial[2*k,k]*Binomial[n+k,n-k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 22 2022 *)

flatten([[(-1)^(n+k)*(2*k+1)*binomial(2*k,k)*binomial(n+k+1,n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Mar 22 2022

T(n, 0) = (-1)^n*A000027(n+1).
T(n, 1) = A331433(n-1) = (-1)^(n+1)*A007531(n+2).
T(n, 2) = A331434(n-2) = (-1)^n*A054559(n+3).
T(n, n-2) = A002738(n-2).
T(n, n-1) = (-1)*A002736(n).
T(n, n) = A002457(n).
T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).
Sum_{k=0..n} T(n, k) = A000290(n+1) (row sums).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).
Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1) (diagonal sums).
2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1) (positive half sums).
(-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n) (negative half sums).
T(n, k) = ((2*k+1)!/(k!)^2)*[x^(n-k)] (1+x)^(-2*(k+1)). - Georg Fischer and Peter Luschny, Jan 18 2020
T(n,k) = (-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!), for n >= k >= 0. - N. J. A. Sloane, Jan 18 2020
From Petros Hadjicostas, Jul 09 2020: (Start)
Michael Somos's formulas above can be restated as
Sum_{k=0..n} T(n,k)/(i+k) = 1 for i = 1..n+1.
These are special cases of the following formula that is alluded to (in some way) in Ser's book:
1 - Sum_{k=0..n} T(n,k)/(x + k) = (x-1)*...*(x-(n + 1))/(x*(x+1)*...*(x+n)).
Because T(n,k) = (-1)^(n+1)*(n + k + 1)*A331430(n,k) and Sum_{k=0..n} A331430(n,k) = (-1)^(n+1), one may derive this formula from Ser's second formula stated in A331430. (End)
T(2*n+1, n) = (-2)*(-27)^n*Pochhammer(4/3, n)*Pochhammer(5/3, n)/(n!*(n+1)!). - G. C. Greubel, Mar 22 2022

Several typos in the data corrected by Georg Fischer and Peter Luschny, Jan 18 2020

Definition changed by N. J. A. Sloane, Jan 18 2020

A137762 Central elements in writing first the numerator and then the denominator (left to right) of Leibniz's harmonic-like triangle.

Original entry on oeis.org

1, 1, 5, 6, 31, 30, 209, 140, 1471, 630, 10625, 2772

Offset: 1

Mohammad K. Azarian, Feb 13 2008

			1/1; --> 1 1
1/2, 1/2; -->
1/3, 5/6, 1/3; --> 5 6
1/4, 7/12, 7/12, 1/4; --> ...
1/5, 9/20, 31/30, 9/20, 1/5;

Cf. A046201; A046204; A046205; A046206; A046208; A046212; A137752; A137753; A137754; A137755; A137756; A137757; A137758; A137759, A137760; A137761.

a(2n-1)=A137763(2n); a(2n)=A137763(2n-1)=A046212(2n-1)=A056040(2n-1)=A100071(2n-1). - M. F. Hasler, Jan 25 2012

A137763 Central elements in writing first the denominator and then the numerator(left to right) of Leibniz's harmonic-like triangle.

Original entry on oeis.org

1, 1, 6, 5, 30, 31, 140, 209, 630, 1471, 2772, 10625

Offset: 1

Mohammad K. Azarian, Feb 13 2008

			1/1; --> 1 1
1/2, 1/2; -->
1/3, 5/6, 1/3;  --> 6 5
1/4, 7/12, 7/12, 1/4; --> ...
1/5, 9/20, 31/30, 9/20, 1/5;

Cf. A046201, A046204, A046205, A046206, A046208, A046212, A137752, A137753, A137754, A137755, A137756, A137757, A137758, A137759, A137760, A137761, A137762.

a(2n) = A137762(2n-1); a(2n-1) = A137762(2n) = A046212(2n-1) = A056040(2n-1) = A100071(2n-1). - M. F. Hasler, Jan 25 2012

A107230 A number triangle of inverse Chebyshev transforms.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 6, 12, 12, 4, 1, 10, 30, 30, 20, 5, 1, 20, 60, 90, 60, 30, 6, 1, 35, 140, 210, 210, 105, 42, 7, 1, 70, 280, 560, 560, 420, 168, 56, 8, 1, 126, 630, 1260, 1680, 1260, 756, 252, 72, 9, 1, 252, 1260, 3150, 4200, 4200, 2520, 1260, 360, 90, 10, 1

Offset: 0

Paul Barry, May 13 2005

First column is A001405, second column is A100071, third column is A107231. Row sums are A005773(n+1), diagonal sums are A026003. The inverse Chebyshev transform concerned takes a g.f. g(x)->(1/sqrt(1-4x^2))g(xc(x^2)) where c(x) is the g.f. of A000108. It transforms a(n) to b(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*a(n-2k). Then a(n) = Sum_{k=0..floor(n/2)} (n/(n-k))*(-1)^k*binomial(n-k,k) *b(n-2k).

Triangle read by rows: T(n,k) is the number of paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., left factors of Motzkin paths) and having k H steps. Example: T(3,1)=6 because we have HUD. HUU, UDH, UHD, UHU and UUH. Sum_{k=0..n} k*T(n,k) = A132894(n). - Emeric Deutsch, Oct 07 2007

			Triangle begins
   1;
   1,  1;
   2,  2,  1;
   3,  6,  3,  1;
   6, 12, 12,  4,  1;
  10, 30, 30, 20,  5,  1;

Jinyuan Wang, Rows n=0..200 of triangle, flattened
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.

Cf. A132894.

[[Binomial(n, k)*Binomial(n-k, Floor((n-k)/2)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 11 2019

T:=proc(n,k) options operator, arrow: binomial(n, k)*binomial(n-k, floor((1/2)*n-(1/2)*k)) end proc: for n from 0 to 11 do seq(T(n,k),k=0..n) end do; # yields sequence in triangular form - Emeric Deutsch, Oct 07 2007

Table[Binomial[n, k]*Binomial[n-k, Floor[(n-k)/2]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2019 *)

T(n, k) = binomial(n, k)*binomial(n-k, (n-k)\2); \\ Michel Marcus, Feb 10 2019

[[binomial(n, k)*binomial(n-k, floor((n-k)/2)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 11 2019

T(n,k) = binomial(n,k)*binomial(n-k, floor((n-k)/2)).
G.f.: G=G(t,z) satisfies z*(1-2*z-t*z)*G^2+(1-2*z-t*z)*G-1=0. - Emeric Deutsch, Oct 07 2007
E.g.f.: exp(x*y)*(BesselI(0,2*x)+BesselI(1,2*x)). - Vladeta Jovovic, Dec 02 2008

cp's OEIS Frontend

A134757 A007318 * A100071.

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A215619 a(n) is the number of consecutive terms of A100071, beginning with index n, which are divisible by n.

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A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).

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A085750 Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

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A046212 First numerator and then denominator of central elements of Leibniz's Harmonic Triangle.

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A037965 a(n) = n*binomial(2*n-2, n-1).

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A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)*(n+k+1)*binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

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A037965 a(n) = nbinomial(2n-2, n-1).

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.