A082505 -id:A082505 - cp's OEIS frontend

A106566 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 14, 14, 9, 4, 1, 0, 42, 42, 28, 14, 5, 1, 0, 132, 132, 90, 48, 20, 6, 1, 0, 429, 429, 297, 165, 75, 27, 7, 1, 0, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1, 0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1

Offset: 0

Philippe Deléham, May 30 2005

Catalan convolution triangle; g.f. for column k: (x*c(x))^k with c(x) g.f. for A000108 (Catalan numbers).
Riordan array (1, xc(x)), where c(x) the g.f. of A000108; inverse of Riordan array (1, x*(1-x)) (see A109466).
Diagonal sums give A132364. - Philippe Deléham, Nov 11 2007

			Triangle begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,  1;
  0,   5,   5,  3,  1;
  0,  14,  14,  9,  4,  1;
  0,  42,  42, 28, 14,  5, 1;
  0, 132, 132, 90, 48, 20, 6, 1;
From _Paul Barry_, Sep 28 2009: (Start)
Production array is
  0, 1,
  0, 1, 1,
  0, 1, 1, 1,
  0, 1, 1, 1, 1,
  0, 1, 1, 1, 1, 1,
  0, 1, 1, 1, 1, 1, 1,
  0, 1, 1, 1, 1, 1, 1, 1,
  0, 1, 1, 1, 1, 1, 1, 1, 1,
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)

Alois P. Heinz, Rows n = 0..140, flattened
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
FindStat - Combinatorial Statistic Finder, The number of touch points of a Dyck path, The number of initial rises of a Dyck paths, The number of nodes on the left branch of the tree, The number of subtrees.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.

Column k for k = 0, 1, 2, ..., 13: A000007, A000108, A000108, A000245, A002057, A000344, A003517, A000588, A003517, A001392, A003518, A000589, A003519, A000590
The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Diagonals: A000012, A001477, A000096, A005586, A005587, A005557, A064059, A064061
See also A009766, A033184, A059365 for other versions.
Generalized Catalan numbers C(x, n) for -11 <= x <= 10: A064333, A064332, A064331, A064330, A064329, A064328, A064327, A064326, A064325, A064311, A064310, A000012, A000108, A064062, A064063, A064087, A064088, A064089, A064090, A064091, A064092, A064093.
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

A106566:= func< n,k | n eq 0 select 1 else (k/n)*Binomial(2*n-k-1, n-k) >;
[A106566(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 06 2021

A106566 := proc(n,k)
    if n = 0 then
        1;
    elif k < 0 or k > n then
        0;
    else
        binomial(2*n-k-1,n-k)*k/n ;
    end if;
end proc: # R. J. Mathar, Mar 01 2015

T[n_, k_] := Binomial[2n-k-1, n-k]*k/n; T[0, 0] = 1; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2017 *)
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, #(1-Sqrt[1-4#])/(2#)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

{T(n, k) = if( k<=0 || k>n, n==0 && k==0, binomial(2*n - k, n) * k/(2*n - k))}; /* Michael Somos, Oct 01 2022 */

def A106566(n, k): return 1 if (n==0) else (k/n)*binomial(2*n-k-1, n-k)
flatten([[A106566(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 06 2021

T(n, k) = binomial(2n-k-1, n-k)*k/n for 0 <= k <= n with n > 0; T(0, 0) = 1; T(0, k) = 0 if k > 0.
T(0, 0) = 1; T(n, 0) = 0 if n > 0; T(0, k) = 0 if k > 0; for k > 0 and n > 0: T(n, k) = Sum_{j>=0} T(n-1, k-1+j).
Sum_{j>=0} T(n+j, 2j) = binomial(2n-1, n), n > 0.
Sum_{j>=0} T(n+j, 2j+1) = binomial(2n-2, n-1), n > 0.
Sum_{k>=0} (-1)^(n+k)*T(n, k) = A064310(n). T(n, k) = (-1)^(n+k)*A099039(n, k).
Sum_{k=0..n} T(n, k)*x^k = A000007(n), A000108(n), A000984(n), A007854(n), A076035(n), A076036(n), A127628(n), A126694(n), A115970(n) for x = 0,1,2,3,4,5,6,7,8 respectively.
Sum_{k>=0} T(n, k)*x^(n-k) = C(x, n); C(x, n) are the generalized Catalan numbers.
Sum_{j=0..n-k} T(n+k,2*k+j) = A039599(n,k).
Sum_{j>=0} T(n,j)*binomial(j,k) = A039599(n,k).
Sum_{k=0..n} T(n,k)*A000108(k) = A127632(n).
Sum_{k=0..n} T(n,k)*(x+1)^k*x^(n-k) = A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x= 0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Aug 25 2007
Sum_{k=0..n} T(n,k)*A000108(k-1) = A121988(n), with A000108(-1)=0. - Philippe Deléham, Aug 27 2007
Sum_{k=0..n} T(n,k)*(-x)^k = A000007(n), A126983(n), A126984(n), A126982(n), A126986(n), A126987(n), A127017(n), A127016(n), A126985(n), A127053(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Oct 27 2007
T(n,k)*2^(n-k) = A110510(n,k); T(n,k)*3^(n-k) = A110518(n,k). - Philippe Deléham, Nov 11 2007
Sum_{k=0..n} T(n,k)*A000045(k) = A109262(n), A000045: Fibonacci numbers. - Philippe Deléham, Oct 28 2008
Sum_{k=0..n} T(n,k)*A000129(k) = A143464(n), A000129: Pell numbers. - Philippe Deléham, Oct 28 2008
Sum_{k=0..n} T(n,k)*A100335(k) = A002450(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A100334(k) = A001906(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A099322(k) = A015565(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A106233(k) = A003462(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A151821(k+1) = A100320(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A082505(k+1) = A144706(n). - Philippe Deléham, Oct 30 2008
Sum_{k=0..n} T(n,k)*A000045(2k+2) = A026671(n). - Philippe Deléham, Feb 11 2009
Sum_{k=0..n} T(n,k)*A122367(k) = A026726(n). - Philippe Deléham, Feb 11 2009
Sum_{k=0..n} T(n,k)*A008619(k) = A000958(n+1). - Philippe Deléham, Nov 15 2009
Sum_{k=0..n} T(n,k)*A027941(k+1) = A026674(n+1). - Philippe Deléham, Feb 01 2014
G.f.: Sum_{n>=0, k>=0} T(n, k)*x^k*z^n = 1/(1 - x*z*c(z)) where c(z) the g.f. of A000108. - Michael Somos, Oct 01 2022

Formula corrected by Philippe Deléham, Oct 31 2008
Corrected by Philippe Deléham, Sep 17 2009
Corrected by Alois P. Heinz, Aug 02 2012

A109466 Riordan array (1, x(1-x)).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 1, -3, 1, 0, 0, 0, 3, -4, 1, 0, 0, 0, -1, 6, -5, 1, 0, 0, 0, 0, -4, 10, -6, 1, 0, 0, 0, 0, 1, -10, 15, -7, 1, 0, 0, 0, 0, 0, 5, -20, 21, -8, 1, 0, 0, 0, 0, 0, -1, 15, -35, 28, -9, 1, 0, 0, 0, 0, 0, 0, -6, 35, -56, 36, -10, 1, 0, 0, 0, 0, 0, 0, 1, -21, 70, -84, 45, -11, 1, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Aug 28 2005

Inverse is Riordan array (1, xc(x)) (A106566).
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008
Coefficient array of the polynomials Chebyshev_U(n, sqrt(x)/2)*(sqrt(x))^n. - Paul Barry, Sep 28 2009

			Rows begin:
  1;
  0,  1;
  0, -1,  1;
  0,  0, -2,  1;
  0,  0,  1, -3,  1;
  0,  0,  0,  3, -4,   1;
  0,  0,  0, -1,  6,  -5,   1;
  0,  0,  0,  0, -4,  10,  -6,   1;
  0,  0,  0,  0,  1, -10,  15,  -7,  1;
  0,  0,  0,  0,  0,   5, -20,  21, -8,  1;
  0,  0,  0,  0,  0,  -1,  15, -35, 28, -9, 1;
From _Paul Barry_, Sep 28 2009: (Start)
Production array is
  0,    1,
  0,   -1,    1,
  0,   -1,   -1,   1,
  0,   -2,   -1,  -1,   1,
  0,   -5,   -2,  -1,  -1,  1,
  0,  -14,   -5,  -2,  -1, -1,  1,
  0,  -42,  -14,  -5,  -2, -1, -1,  1,
  0, -132,  -42, -14,  -5, -2, -1, -1,  1,
  0, -429, -132, -42, -14, -5, -2, -1, -1, 1 (End)

Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
Tom Copeland, Addendum to Elliptic Lie Triad

Cf. A026729 (unsigned version), A000108, A030528, A124644.

/* As triangle */ [[(-1)^(n-k)*Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 14 2016

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, #(1-#)&, 13] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Number triangle T(n, k) = (-1)^(n-k)*binomial(k, n-k).
T(n, k)*2^(n-k) = A110509(n, k); T(n, k)*3^(n-k) = A110517(n, k).
Sum_{k=0..n} T(n,k)*A000108(k)=1. - Philippe Deléham, Jun 11 2007
From Philippe Deléham, Oct 30 2008: (Start)
Sum_{k=0..n} T(n,k)*A144706(k) = A082505(n+1).
Sum_{k=0..n} T(n,k)*A002450(k) = A100335(n).
Sum_{k=0..n} T(n,k)*A001906(k) = A100334(n).
Sum_{k=0..n} T(n,k)*A015565(k) = A099322(n).
Sum_{k=0..n} T(n,k)*A003462(k) = A106233(n). (End)
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = -12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 respectively. - Philippe Deléham, Oct 27 2008
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n+1), A057086(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively. - Philippe Deléham, Oct 28 2008
G.f.: 1/(1-y*x+y*x^2). - Philippe Deléham, Dec 15 2011
T(n,k) = T(n-1,k-1) - T(n-2,k-1), T(n,0) = 0^n. - Philippe Deléham, Feb 15 2012
Sum_{k=0..n} T(n,k)*x^(n-k) = F(n+1,-x) where F(n,x)is the n-th Fibonacci polynomial in x defined in A011973. - Philippe Deléham, Feb 22 2013
Sum_{k=0..n} T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 26 2013
Sum_{k=0..n} T(n,k)*T(n+1,k) = -A110320(n). - Philippe Deléham, Feb 26 2013
For T(0,0) = 0, the signed triangle below has the o.g.f. G(x,t) = [t*x(1-x)]/[1-t*x(1-x)] = L[t*Cinv(x)] where L(x) = x/(1-x) and Cinv(x)=x(1-x) with the inverses Linv(x) = x/(1+x) and C(x)= [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108, so the inverse o.g.f. is Ginv(x,t) = C[Linv(x)/t] = [1-sqrt[1-4*x/(t(1+x))]]/2 (cf. A124644 and A030528). - Tom Copeland, Jan 19 2016

A093562 (5,1) Pascal triangle.

Original entry on oeis.org

1, 5, 1, 5, 6, 1, 5, 11, 7, 1, 5, 16, 18, 8, 1, 5, 21, 34, 26, 9, 1, 5, 26, 55, 60, 35, 10, 1, 5, 31, 81, 115, 95, 45, 11, 1, 5, 36, 112, 196, 210, 140, 56, 12, 1, 5, 41, 148, 308, 406, 350, 196, 68, 13, 1, 5, 46, 189, 456, 714, 756, 546, 264, 81, 14, 1, 5, 51, 235, 645, 1170

Offset: 0

Wolfdieter Lang, Apr 22 2004

This is the fifth member, d=5, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-1, for d=1..4.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=(1+4*z)/(1-(1+x)*z).
The SW-NE diagonals give A022095(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 4. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
The array F(5;n,m) gives in the columns m >= 1 the figurate numbers based on A016861, including the heptagonal numbers A000566 (see the W. Lang link).
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
The n-th row polynomial is (4 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Mar 02 2018

			Triangle begins
  [1];
  [5,  1];
  [5,  6,  1];
  [5, 11,  7,  1];
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
P. Bala, A note on the diagonals of a proper Riordan Array
W. Lang, First 10 rows and array of figurate numbers .

Cf. Row sums: A007283(n-1), n>=1, 1 for n=0. A082505(n+1), alternating row sums are 1 for n=0, 4 for n=2 and 0 else.
Column sequences give for m=1..9: A016861, A000566 (heptagonal), A002413, A002418, A027800, A051946, A050484, A052255, A055844.
A007318, A093563 (d=6), A228196, A228576.

a093562 n k = a093562_tabl !! n !! k
a093562_row n = a093562_tabl !! n
a093562_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [5, 1]
-- Reinhard Zumkeller, Aug 31 2014

from math import comb, isqrt
def A093562(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*(r+(r-a<<2))//r if n else 1 # Chai Wah Wu, Nov 12 2024

a(n, m) = F(5;n-m, m) for 0<= m <= n, otherwise 0, with F(5;0, 0)=1, F(5;n, 0)=5 if n>=1 and F(5;n, m):=(5*n+m)*binomial(n+m-1, m-1)/m if m>=1.
G.f. column m (without leading zeros): (1+4*x)/(1-x)^(m+1), m>=0.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=5 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
T(n, k) = C(n, k) + 4*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(5 + 11*x + 7*x^2/2! + x^3/3!) = 5 + 16*x + 34*x^2/2! + 60*x^3/3! + 95*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A048488 a(n) = 6*2^n - 5.

Original entry on oeis.org

1, 7, 19, 43, 91, 187, 379, 763, 1531, 3067, 6139, 12283, 24571, 49147, 98299, 196603, 393211, 786427, 1572859, 3145723, 6291451, 12582907, 25165819, 50331643, 100663291, 201326587, 402653179, 805306363, 1610612731

Offset: 0

Clark Kimberling, Dec 11 1999

a(n) = T(5, n), array T given by A048483.

Sequence is generated by the Northwest (NW) direction of circles put around circle(s). See illustration. - Odimar Fabeny, Aug 09 2008

			a(2) = 6 * 2^2 - 5 = 6 * 4 - 5 = 24 - 5 = 19.
a(3) = 6 * 2^3 - 5 = 6 * 8 - 5 = 48 - 5 = 43.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Odimar Fabeny, Illustration for this sequence
Index entries for linear recurrences with constant coefficients, signature (3,-2).

n-th difference of a(n), a(n-1), ..., a(0) is (6, 6, 6, ...).

Cf. A000079, A007283. - Omar E. Pol, Dec 21 2008

[6*2^n - 5: n in [0..30]]; // Vincenzo Librandi, May 18 2011

A048488:=n->6*2^n - 5; seq(A048488(n), n=0..30); # Wesley Ivan Hurt, May 09 2014

6(2^Range[0, 35]) - 5 (* Alonso del Arte, May 03 2014 *)

a(n)=6*2^n-5 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*a(n-1) + 5, n > 0, a(0) = 1. - Paul Barry, Aug 25 2004
Equals binomial transform of [1, 6, 6, 6, ...]. - Gary W. Adamson, Apr 29 2008
a(n) = A000079(n)*6 - 5 = A007283(n)*2 - 5. - Omar E. Pol, Dec 21 2008
From Colin Barker, Sep 17 2012: (Start)
a(n) = 3*2^(1+n) - 5. a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: (1+4*x)/((1-x)*(1-2*x)). (End)
a(n + 1) = 3 * 2^n - 5 = 1 + 2 * (Sum_{i=0..n-1} 3i) for n > 0. - Gerasimov Sergey and Alonso del Arte, May 03 2014
a(n) = A000225(n+1)+4*A000225(n). - R. J. Mathar, Feb 27 2019

Simpler definition from Ralf Stephan

A110164 Expansion of (1-x^2)/(1+2x).

Original entry on oeis.org

1, -2, 3, -6, 12, -24, 48, -96, 192, -384, 768, -1536, 3072, -6144, 12288, -24576, 49152, -98304, 196608, -393216, 786432, -1572864, 3145728, -6291456, 12582912, -25165824, 50331648, -100663296, 201326592, -402653184, 805306368, -1610612736, 3221225472

Offset: 0

Paul Barry, Jul 14 2005

Diagonal sums of Riordan array ((1-x)/(1+x),x/(1+x)^2), A110162.
The positive sequence with g.f. (1-x^2)/(1-2x) gives the row sums of the Riordan array (1+x,x/(1-x)). - Paul Barry, Jul 18 2005
The inverse g.f. is (1 + 2*x + x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 + ...). - Gary W. Adamson, Jan 07 2011
In absolute value, essentially the same as A007283(n) = A003945(n+1) = A042950(n+1) = A082505(n+1) = A087009(n+3) = A091629(n) = A098011(n+4) = A111286(n+2). - M. F. Hasler, Apr 19 2015

Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A098011, A122803.

CoefficientList[Series[(1 - x^2)/(1 + 2x), {x, 0, 33}], x] (* Robert G. Wilson v, Jul 08 2006 *)
LinearRecurrence[{-2},{1,-2,3},40] (* Harvey P. Dale, May 10 2023 *)

A110164(n)=if(n>1,3*(-2)^(n-2),1-3*n) \\ M. F. Hasler, Apr 19 2015

a(n) = 3*(-2)^(n-2) = 3*A122803(n-2) for n >= 2. a(n) = -2 a(n-1) for n >= 3. - M. F. Hasler, Apr 19 2015

E.g.f.: (1/4) - (x/2) + (3/4)*exp(-2*x). - Alejandro J. Becerra Jr., Jan 29 2021

A144706 Central coefficients of the triangle A132047.

Original entry on oeis.org

1, 6, 18, 60, 210, 756, 2772, 10296, 38610, 145860, 554268, 2116296, 8112468, 31201800, 120349800, 465352560, 1803241170, 7000818660, 27225405900, 106035791400, 413539586460, 1614773623320, 6312296891160, 24700292182800, 96742811049300, 379231819313256

Offset: 0

Paul Barry, Sep 19 2008

Hankel transform is A144708.

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

Cf. A000108, A010686, A039599, A082505, A106566, A132047, A144708.

[n eq 0 select 1 else 3*(n+1)*Catalan(n): n in [0..40]]; // G. C. Greubel, Jun 16 2022

Table[3*Binomial[2n,n] -2*Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 16 2022 *)

a(n) = if(n, 3*binomial(2*n, n), 1) \\ Charles R Greathouse IV, Oct 23 2023

[3*binomial(2*n, n) -2*bool(n==0) for n in (0..40)] # G. C. Greubel, Jun 16 2022

G.f.: 3/sqrt(1-4*x) - 2;
a(n) = 3*binomial(2*n, n) - 2*0^n.
From Philippe Deléham, Oct 30 2008: (Start)
a(n) = Sum_{k=0..n} A039599(n,k)*A010686(k).
a(n) = Sum_{k=0..n} A106566(n,k)*A082505(k+1). (End)
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1). - R. J. Mathar, Nov 30 2012
E.g.f.: -2 + 3*exp(2*x)*BesselI(0, 2*x). - G. C. Greubel, Jun 16 2022

A047225 Numbers that are congruent to {0, 1} mod 6.

Original entry on oeis.org

0, 1, 6, 7, 12, 13, 18, 19, 24, 25, 30, 31, 36, 37, 42, 43, 48, 49, 54, 55, 60, 61, 66, 67, 72, 73, 78, 79, 84, 85, 90, 91, 96, 97, 102, 103, 108, 109, 114, 115, 120, 121, 126, 127, 132, 133, 138, 139, 144, 145, 150

Offset: 1

N. J. A. Sloane

Also: 0 followed by partial sums of A010686. - R. J. Mathar, Feb 23 2008
Expansion of 1/(1 + x + x^2 + x^3 + x^4 + x^5) = 1 - x + x^6 - x^7 + x^12 - x^13 + ... and the exponents are the terms of this sequence. - Gary W. Adamson, Apr 04 2011
Numbers k such that floor(k/2) = 3*floor(k/6). - Bruno Berselli, Oct 05 2017

David Lovler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A010686, A030308, A082505.

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+6 od: seq(a[n], n=0..50); # Zerinvary Lajos, Mar 16 2008

{#,#+1}&/@(6Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{0,1,6},60] (* Harvey P. Dale, Aug 24 2019 *)

forstep(n=0,200,[1,5],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

From R. J. Mathar, Feb 23 2008: (Start)
O.g.f.: 1/(1+x) + 3/(-1+x)^2 + 4/(-1+x).
a(n) = a(n-2) + 6, n >= 2. (End)
a(n) = 6*n - a(n-1) - 11 for n>1, a(1)=0. - Vincenzo Librandi, Aug 05 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*A082505(k+1). - Philippe Deléham, Oct 17 2011
Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/12 + log(2)/3 + log(3)/4. - Amiram Eldar, Dec 13 2021
E.g.f.: 5 + (3*x - 4)*exp(x) - exp(-x). - David Lovler, Aug 25 2022

Formula corrected by Paolo P. Lava, Oct 12 2010

A134059 Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.

Original entry on oeis.org

1, 3, 3, 3, 6, 3, 3, 9, 9, 3, 3, 12, 18, 12, 3, 3, 15, 30, 30, 15, 3, 3, 18, 45, 60, 45, 18, 3, 3, 21, 63, 105, 105, 63, 21, 3, 3, 24, 84, 168, 210, 168, 84, 24, 3, 3, 27, 108, 252, 378, 378, 252, 108, 27, 3

Offset: 0

Gary W. Adamson, Oct 05 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [3, -2, 0, 0, 0, 0, 0, ...] DELTA [3, -2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007

			First few rows of the triangle:
  1;
  3,  3;
  3,  6,  3;
  3,  9,  9,  3;
  3, 12, 18, 12,  3;
  3, 15, 30, 30, 15,  3;
  3, 18, 45, 60, 45, 18,  3;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5150 (rows n = 0..100, flattened)

Cf. A007318, A082505 (row sums), A084938, A134058.

A134059:= func< n,k | n eq 0 select 1 else 3*Binomial(n,k) >;
[A134059(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2021

Join[{1},Rest[Flatten[Table[3Binomial[n,k],{n,0,10},{k,0,n}]]]] (* Harvey P. Dale, Feb 15 2014 *)
Table[3*Binomial[n,k] -2*Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 26 2021 *)

def A134059(n,k): return 3*binomial(n,k) - 2*bool(n==0)
flatten([[A134059(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2021

3*Pascal's triangle A007318, then replace T(0,0) with 1.
G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k) *x^n * y^k = 1 - 3*(1+y)*x/(x+x*y-1). - R. J. Mathar, Feb 19 2020
From G. C. Greubel, Apr 27 2021: (Start)
T(n, k) = 3*binomial(n,k) - 2*[n=0].
Sum_{k=0..n} T(n, k) = 3*2^n - 2*[n=0] = A082505(n+1). (End)
E.g.f.: 3*exp(x*(1+y)) - 2. - Stefano Spezia, Apr 03 2024

Title changed by G. C. Greubel, Apr 26 2021

A165665 a(n) = (32^n - 2) 2^n.

Original entry on oeis.org

1, 8, 40, 176, 736, 3008, 12160, 48896, 196096, 785408, 3143680, 12578816, 50323456, 201310208, 805273600, 3221159936, 12884770816, 51539345408, 206157905920, 824632672256, 3298532786176, 13194135339008, 52776549744640

Offset: 0

Klaus Brockhaus, Sep 24 2009

Binomial transform of A058481. Second binomial transform of (A082505 without initial term 0). Third binomial transform of A010686.
Partial sums are in A060867.
a(n) is the sum of the odd numbers taken progressively by moving through them by 2^n-tuples. a(0)=1; a(1) = 3+5=8; a(2) = 7+9+11+13 = 40; a(3) = 15+17+19+21+23+25+27+29 = 176; a(n) = sum_{k=0,1,..,A000225(n)} (A000225(n+1)+2*k).  - J. M. Bergot, Dec 06 2014
The number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A058481, A082505, A010686 (repeat 1, 5), A060867, A010036, A124647.

Magma
```
[ (3*2^n-2)*2^n: n in [0..23] ];
```

Table[(3*2^n-2)2^n,{n,0,30}] (* or  *) LinearRecurrence[{6,-8},{1,8},30] (* Harvey P. Dale, Nov 18 2020 *)

a(n)=(3*2^n-2)*2^n \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 6*a(n-1)-8*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
a(n) = 8*A010036(n-1) for n > 0.
G.f.: (2*x+1)/((1-2*x)*(1-4*x)).
E.g.f.: 3*e^(4*x) - 2*e^(2*x). - Robert Israel, Dec 15 2014

cp's OEIS Frontend

A106566 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938.

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A109466 Riordan array (1, x(1-x)).

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A093562 (5,1) Pascal triangle.

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A048488 a(n) = 6*2^n - 5.

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A110164 Expansion of (1-x^2)/(1+2x).

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A144706 Central coefficients of the triangle A132047.

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A165665 a(n) = (32^n - 2) 2^n.