A001792 -id:A001792 - cp's OEIS frontend

A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

1, 0, 3, 2, 9, 12, 31, 54, 117, 224, 459, 906, 1825, 3636, 7287, 14558, 29133, 58248, 116515, 233010, 466041, 932060, 1864143, 3728262, 7456549, 14913072, 29826171, 59652314, 119304657, 238609284, 477218599, 954437166, 1908874365

Offset: 0

Pauline Gorman (pauline(AT)gorman65.freeserve.co.uk), Feb 26 2000

Growth of happy bug population in GCSE math course work assignment.
The generalized (3,2)-Padovan sequence p(3,2;n). See the W. Lang link under A000931. - Wolfdieter Lang, Jun 25 2010
With offset 1: a(n) = -2^n*Sum_{k=0..n} k^p*q^k for p=1, q=-1/2. See also A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2). - Stanislav Sykora, Nov 27 2013
From Paul Curtz, Nov 02 2021 (Start)
a(n-2) difference table (from 0, 0, a(n)):
    0    0    1    0    3    2    9    12    31    54  ...
    0    1   -1    3   -1    7    3    19    23    63  ...
    1   -2    4   -4    8   -4   16     4    40    44  ...
   -3    6   -8   12  -12   20  -12    36     4    84  ...
    9  -14   20  -24   32  -32   48   -32    80     0  ...
  -23   34  -44   56  -64   80  -80   112   -80   176  ...
   57  -78  100 -120  144 -160  192  -192   256  -192  ...
   ... .
The signature is valid for every row.
a(n-2) + a(n-1) = A001045(n).
a(n-2) + a(n+1) = A062510(n) = 3*A001045(n).
a(n-2) + a(n+3) = see A144472(n+1).
Second subdiagonal: 1, 6, 20, 56, 144, 352, ... = A014480(n).
First subdiagonal: -A036895(n) = -2*A001787(n).
Main diagonal: A001787(n) = -first and -third upper diagonals.
Second, fourth and fifth upper diagonals: A001792(n), A045891(n+2) and A172160(n+1). (End)

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Iwan Duursma, Xiao Li, and Hsin-Po Wang, Multilinear Algebra for Distributed Storage, arXiv:2006.08911 [cs.IT], 2020.
Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Cf. A232603, A232604.

Cf. A001045, A001787, A001792, A014480, A036895, A062510, A045891, A172160.

CoefficientList[Series[1/(1 - 3 x^2 - 2 x^3), {x, 0, 32}], x] (* Michael De Vlieger, Sep 30 2019 *)

c(n)=(2^(n+1)-(-1)^n*(3*n+2))/9; a(n)=c(n+1); \\ Stanislav Sykora, Nov 27 2013

G.f.: 1 / (1-3*x^2-2*x^3).
With offset 1: a(1)=1; a(n) = 2*a(n-1) - (-1)^n*n; a(n) = (1/9)*(2^(n+1) - (-1)^n*(3*n+2)). - Benoit Cloitre, Nov 02 2002
a(n) = Sum_{k=0..floor(n/2)} A078008(n-2k). - Paul Barry, Nov 24 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)*3^k*(2/3)^(n-2k). - Paul Barry, Oct 16 2004
a(n) = Sum_{k=0..n} A078008(k)*(1 - (-1)^(n+k-1))/2. - Paul Barry, Apr 16 2005
a(n) = ( 2^(n+2) + (-1)^n*(3*n+5) )/9 (see also the B. Cloitre comment above). From the o.g.f. 1/(1-3*x^2-2*x^3) = 1/((1-2*x)*(1+x)^2) = (3/(1+x)^2 + 2/(1+x) + 4/(1-2*x))/9. - Wolfdieter Lang, Jun 25 2010
From Wolfdieter Lang, Aug 26 2010: (Start)
a(n) = a(n-1) + 2*a(n-2) + (-1)^n for n > 1, a(0)=1, a(1)=0.
Due to the identity for the o.g.f. A(x): A(x) = x*(1+2*x)*A(x) + 1/(1+x).
(This recurrence was observed by Gary Detlefs in a 08/25/10 e-mail to the author.) (End)
G.f.: Sum_{n>=0} binomial(3*n,n)*x^n / (1+x)^(3*n+3). - Paul D. Hanna, Mar 03 2012
E.g.f.: 1 + (1/9)*(exp(-x)*(3*x - 2) + 2*exp(2*x)). - Stefano Spezia, Sep 27 2019

More terms from James Sellers, Feb 28 2000 and Christian G. Bower, Feb 29 2000

A053469 a(n) = n*6^(n-1).

Original entry on oeis.org

1, 12, 108, 864, 6480, 46656, 326592, 2239488, 15116544, 100776960, 665127936, 4353564672, 28298170368, 182849716224, 1175462461440, 7522959753216, 47958868426752, 304679870005248, 1929639176699904, 12187194800209920, 76779327241322496, 482612914088312832

Offset: 1

Barry E. Williams, Jan 13 2000

Binomial transform of A053464. - R. J. Mathar, Oct 26 2011

			G.f. = x + 12*x^2 + 108*x^3 + 864*x^4 + 6480*x^5 + 46656*x^6 + ... - _Michael Somos_, Dec 16 2019

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 1..400
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A002697, A027471.

[n*(6^(n-1)): n in [1..30]]; // Vincenzo Librandi, Jun 09 2011

f[n_]:=n*6^(n-1);f[Range[40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{12,-36},{1,12},20] (* Harvey P. Dale, Apr 28 2015 *)

a(n)=n*6^(n-1) \\ Charles R Greathouse IV, Oct 07 2015

[lucas_number1(n,12,36) for n in range(1, 21)] # Zerinvary Lajos, Apr 28 2009

a(n) = 12*a(n-1) - 36*a(n-2), n>=3.
G.f.: x/(6x-1)^2. - Zerinvary Lajos, Apr 28 2009
E.g.f.: x*exp(6*x). - Michael Somos, Dec 16 2019
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 6*log(6/5).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(7/6). (End)

More terms from James Sellers, Feb 02 2000

More terms from Zerinvary Lajos, Oct 02 2007

A059576 Summatory Pascal triangle T(n,k) (0 <= k <= n) read by rows. Top entry is 1. Each entry is the sum of the parallelogram above it.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 8, 8, 4, 8, 20, 26, 20, 8, 16, 48, 76, 76, 48, 16, 32, 112, 208, 252, 208, 112, 32, 64, 256, 544, 768, 768, 544, 256, 64, 128, 576, 1376, 2208, 2568, 2208, 1376, 576, 128, 256, 1280, 3392, 6080, 8016, 8016, 6080, 3392, 1280, 256

Offset: 0

Floor van Lamoen, Jan 23 2001

We may also relabel the entries as U(0,0), U(1,0), U(0,1), U(2,0), U(1,1), U(0,2), U(3,0), ... [That is, T(n,k) = U(n-k, k) for 0 <= k <= n and U(m,s) = T(m+s, s) for m,s >= 0.]
From Petros Hadjicostas, Jul 16 2020: (Start)
We explain the parallelogram definition of T(n,k).
    T(0,0) *
           |\
           | \
           |  * T(k,k)
  T(n-k,0) *  |
            \ |
             \|
              * T(n,k)
The definition implies that T(n,k) is the sum of all T(i,j) such that (i,j) has integer coordinates over the set
{(i,j): a(1,0) + b(1,1), 0 <= a <= n-k, 0 <= b <= k} - {(n,k)}.
The parallelogram can sometimes be degenerate; e.g., when k = 0 or n = k. (End)
T(n,k) is the number of 2-compositions of n having sum of the entries of the first row equal to k (0 <= k <= n). A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n. - Emeric Deutsch, Oct 12 2010
From Michel Marcus and Petros Hadjicostas, Jul 16 2020: (Start)
Robeva and Sun (2020) let A(m,n) = U(m-1, n-1) be the number of subdivisions of a 2-row grid with m points on the top and n points at the bottom (and such that the lower left point is the origin).
The authors proved that A(m,n) = 2*(A(m,n-1) + A(m-1,n) - A(m-1,n-1)) for m, n >= 2 (with (m,n) <> (2,2)), which is equivalent to a similar recurrence for U(n,k) given in the Formula section below. (They did not explicitly specify the value of A(1,1) = U(0,0) because they did not care about the number of subdivisions of a degenerate polygon with only one side.)
They also proved that, for (m,n) <> (1,1), A(m,n) = (2^(m-2)/(n-1)!) * Q_n(m) =
= (2^(m-2)/(n-1)!) * Sum_{k=1..n} A336244(n,k) * m^(n-k), where Q_n(m) is a polynomial in m of degree n-1. (End)
With the square array notation of Petros Hadjicostas, Jul 16 2020 below, U(i,j) is the number of lattice paths from (0,0) to (i,j) whose steps move north or east or have positive slope. For example, representing a path by its successive lattice points rather than its steps, U(1,2) = 8 counts {(0,0),(1,2)}, {(0,0),(0,1),(1,2)}, {(0,0),(0,2),(1,2)}, {(0,0),(1,0),(1,2)}, {(0,0),(1,1),(1,2)}, {(0,0),(0,1),(0,2),(1,2)}, {(0,0),(0,1),(1,1),(1,2)}, {(0,0),(1,0),(1,1),(1,2)}. If north (vertical) steps are excluded, the resulting paths are counted by A049600.  - David Callan, Nov 25 2021

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins
[0]   1;
[1]   1,   1;
[2]   2,   3,   2;
[3]   4,   8,   8,   4;
[4]   8,  20,  26,  20,   8;
[5]  16,  48,  76,  76,  48,  16;
[6]  32, 112, 208, 252, 208, 112, 32;
  ...
T(5,2) = 76 is the sum of the elements above it in the parallelogram bordered by T(0,0), T(5-2,0) = T(3,0), T(2,2) and T(5,2). We of course exclude T(5,2) from the summation. Thus
T(5,2) = Sum_{a=0..5-2, b=0..2, (a,b) <> (5-2,2)} T(a(1,0) + b(1,1)) =
= (1 + 1 + 2) + (1 + 3 + 8) + (2 + 8 + 26) + (4 + 20) = 76. [Edited by _Petros Hadjicostas_, Jul 16 2020]
From _Petros Hadjicostas_, Jul 16 2020: (Start)
Square array U(n,k) (with rows n >= 0 and columns k >= 0) begins
   1,   1,   2,    4,    8, ...
   1,   3,   8,   20,   48, ...
   2,   8,  26,   76,  208, ...
   4,  20,  76,  252,  768, ...
   8,  48, 208,  768, 2568, ...
  16, 112, 544, 2208, 8016, ...
  ...
Consider the following 2-row grid with n = 3 points at the top and k = 2 points at the bottom:
   A  B  C
   *--*--*
   |    /
   |   /
   *--*
   D  E
The sets of the dividing internal lines of the A(3,2) = U(3-1, 2-1) = 8 subdivisions of the above 2-row grid are as follows: { }, {DC}, {DB}, {EB}, {EA}, {DB, DC}, {DB, EB}, and {EA, EB}. See Robeva and Sun (2020).
These are the 2-compositions of n = 3 with sum of first row entries equal to k = 1:
[1; 2], [0,1; 2,0], [0,1; 1,1], [1,0; 0,2], [1,0; 1,1], [0,0,1; 1,1,0], [0,1,0; 1,0,1], and [1,0,0; 0,1,1]. We have T(3,2) = 8 such matrices. See _Emeric Deutsch_'s contribution above. See also Section 2 in Castiglione et al. (2007). (End)

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28(6) (2007), 1724-1741; see Fig. 5, p. 1729.
Fang, E., Jenkins, J., Lee, Z., Li, D., Lu, E., Miller, S. J., ... & Siktar, J. (2019). Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice, arXiv preprint arXiv:1906.10645. Also Fib. Q., 58:1 (2020), 208-225.
Elina Robeva and Melinda Sun, Bimonotone Subdivisions of Point Configurations in the Plane, arXiv:2007.00877 [math.CO], 2020.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000079, A001792, A003480 (row sums), A052141 (main diagonal).
Cf. A059474, A059473, A008288, A035002, A059226, A059283, A181292, A336244.
Cf. A011782, A049611, A049612, A055589, A055852, A055853, A181306.

a059576 n k = a059576_tabl !! n !! k
a059576_row n = a059576_tabl !! n
a059576_tabl = [1] : map fst (iterate f ([1,1], [2,3,2])) where
   f (us, vs) = (vs, map (* 2) ws) where
     ws = zipWith (-) (zipWith (+) ([0] ++ vs) (vs ++ [0]))
                      ([0] ++ us ++ [0])
-- Reinhard Zumkeller, Dec 03 2012

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function T(n,k) // T = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*T(n-1, k-1) + 2*T(n-1, k) - (2 - 0^(n-2))*T(n-2, k-1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 02 2022

A059576 := proc(n,k) local b,t1; t1 := min(n+k-2,n,k); add( (-1)^b * 2^(n+k-b-2) * (n+k-b-2)! * (1/(b! * (n-b)! * (k-b)!)) * (-2 * n-2 * k+2 * k^2+b^2-3 * k * b+2 * n^2+5 * n * k-3 * n * b), b=0..t1); end;
T := proc (n, k) if k <= n then add((-1)^j*2^(n-j-1)*binomial(k, j)*binomial(n-j, k), j = 0 .. min(k, n-k)) fi end proc: 1; for n to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form # Emeric Deutsch, Oct 12 2010
T := (n, k) -> `if`(n=0, 1, 2^(n-1)*binomial(n, k)*hypergeom([-k, k - n], [-n], 1/2)): seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Nov 26 2021

T[0, 0] = 1; T[n_, k_] := 2^(n-k-1)*n!*Hypergeometric2F1[ -k, -k, -n, -1 ] / (k!*(n-k)!); Flatten[ Table[ T[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Feb 01 2012, after Robert Israel *)

def T(n,k): # T = A059576
    if (k==0 or k==n): return 1 if (n==0) else 2^(n-1) # A011782
    else: return 2*T(n-1, k-1) + 2*T(n-1, k) - (2 - 0^(n-2))*T(n-2, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 02 2022

T(n, n-1) = A001792(n-1).
T(2*n, n) = A052141(n).
Sum_{k=0..n} T(n, k) = A003480(n).
G.f.: U(z, w) = Sum_{n >= 0, k >= 0} U(n, k)*z^n*w^k = Sum{n >= 0, k >= 0} T(n, k)*z^(n-k)*w^k = (1-z)*(1-w)/(1 - 2*w - 2*z + 2*z*w).
Maple code gives another explicit formula for U(n, k).
From Jon Stadler (jstadler(AT)capital.edu), Apr 30 2003: (Start)
U(n,k) is the number of ways of writing the vector (n,k) as an ordered sum of vectors, equivalently, the number of paths from (0,0) to (n,k) in which steps may be taken from (i,j) to (p,q) provided (p,q) is to the right or above (i,j).
2*U(n,k) = Sum_{i <= n, j <= k} U(i,j).
U(n,k) = 2*U(n-1,k) + Sum_{i < k} U(n,i).
U(n,k) = Sum_{j=0..n+k} C(n,j-k+1)*C(k,j-n+1)*2^j. (End)
T(n, k) = 2*(T(n-1, k-1) + T(n-1, k)) - (2 - 0^(n-2))*T(n-2, k-1) for n > 1 and 1 < k < n; T(n, 0) = T(n, n) = 2*T(n-1, 0) for n > 0; and T(0, 0) = 1. - Reinhard Zumkeller, Dec 03 2004
From Emeric Deutsch, Oct 12 2010: (Start)
Sum_{k=0..n} k*T(n,k) = A181292(n).
T(n,k) = Sum_{j=0..min(k, n-k)} (-1)^j*2^(n-j-1)*binomial(k, j)*binomial(n-j, k) for (n,k) != (0,0).
G.f.: G(t,z) = (1-z)*(1-t*z)/(1 - 2*z - 2*t*z + 2*t*z^2). (End)
U(n,k) = 0 if k < 0; else U(k,n) if k > n; else 1 if n <= 1; else 3 if n = 2 and k = 1; else 2*U(n,k-1) + 2*U(n-1,k) - 2*U(n-1,k-1). - David W. Wilson; corrected in the case k > n by Robert Israel, Jun 15 2011 [Corrected by Petros Hadjicostas, Jul 16 2020]
U(n,k) = binomial(n,k) * 2^(n-1) * hypergeom([-k,-k], [n+1-k], 2) if n >= k >= 0 with (n,k) <> (0,0). - Robert Israel, Jun 15 2011 [Corrected by Petros Hadjicostas, Jul 16 2020]
U(n,k) = Sum_{0 <= i+j <= n+k-1} (-1)^j*C(i+j+1, j)*C(n+i, n)*C(k+i, k). - Masato Maruoka, Dec 10 2019
T(n, k) = 2^(n - 1)*binomial(n, k)*hypergeom([-k, k - n], [-n], 1/2) = A059474(n, k)/2 for n >= 1. - Peter Luschny, Nov 26 2021
From G. C. Greubel, Sep 02 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = T(n, n) = A011782(n).
T(n, n-2) =  2*A049611(n-1), n >= 2.
T(n, n-3) =  4*A049612(n-2), n >= 3.
T(n, n-4) =  8*A055589(n-3), n >= 4.
T(n, n-5) = 16*A055852(n-4), n >= 5.
T(n, n-6) = 32*A055853(n-5), n >= 6.
Sum_{k=0..floor(n/2)} T(n, k) = A181306(n). (End)

A124736 Table where row n has k C(n,k-1) times.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 3, 3, 3, 4, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

			The table starts:
1
1 2
1 2 2 3
1 2 2 2 3 3 3 4

Cf. A124737, A124748, A000079 (row lengths), A001792 (row sums), A007318.

a(n) = A124737(n) + 1

A152920 Triangle read by rows: triangle A062111 reversed.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 3, 5, 8, 12, 4, 7, 12, 20, 32, 5, 9, 16, 28, 48, 80, 6, 11, 20, 36, 64, 112, 192, 7, 13, 24, 44, 80, 144, 256, 448, 8, 15, 28, 52, 96, 176, 320, 576, 1024, 9, 17, 32, 60, 112, 208, 384, 704, 1280, 2304, 10, 19, 36, 68, 128, 240, 448, 832, 1536, 2816, 5120

Offset: 0

Paul Curtz, Dec 15 2008

			Triangle starts:
  0;
  1,  1;
  2,  3,  4;
  3,  5,  8, 12;
  4,  7, 12, 20, 32;
  ...

Alois P. Heinz, Rows n = 0..150, flattened

Cf. A053220, A058877 (row sums), A193844, A212697.

Columns and diagonals: A001787, A001792, A034007, A045623, A045891, A111297, A159694, A159695, A159696, A159697.

[2^k*(n-k/2): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2022

A062111 := proc(n,k) (k+n)*2^(k-n-1) ; end: A152920 := proc(n,k) A062111(n-k,n) ; end: for n from 0 to 15 do for k from 0 to n do printf("%d,",A152920(n,k)) ; od: od: # R. J. Mathar, Jan 22 2009
# second Maple program:
T:= proc(n, k) option remember;
     `if`(k=0, n, T(n, k-1)+T(n-1, k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 12 2022

t[0, k_]:= k; t[n_, k_]:= t[n, k]= t[n-1, k] + t[n-1, k+1];
Table[t[n-k, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Jean-François Alcover, Sep 11 2016 *)

flatten([[2^(k-1)*(2*n-k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Sep 27 2022

Row sums: (2^n-1)(n+1) = A058877(n). - R. J. Mathar, Jan 22 2009
T(2n, n) = 3*n*2^(n-1) = 3*A001787(n). - Philippe Deléham, Apr 20 2009
From Werner Schulte, Jul 31 2020: (Start)
T(n, k) = (2*n-k) * 2^(k-1) for 0 <= k <= n.
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = t*(1+x-3*x*t) / ((1-t)^2 * (1-2*x*t)^2).
Sum_{k=0..n} (-1)^k * binomial(n,k) * T(n,k) = 0 for n >= 0.
Sum_{k=0..n} binomial(n,k) * T(n,k) = 2*n * 3^(n-1) for n >= 0.
Define the array B(n,p) = (Sum_{k=0..n} binomial(p+k,p) * T(n,k))/(n+p+1) for n >= 0 and p >= 0. Then see the comment of Robert Coquereaux (2014) at A193844. Conjecture: B(n+1,p) = A(n,p). (End)
T(n, k) = T(n, k-1) + T(n-1, k-1) for k>=1, T(n,0) = n. - Alois P. Heinz, Sep 12 2022
From G. C. Greubel, Sep 27 2022: (Start)
T(n, n-1) = A001792(n).
T(2*n-1, n-1) = A053220(n).
T(2*n+1, n-1) = 3*A001792(n).
T(m*n, n) = (2*m-1)*A001787(n), for m >= 1. (End)

Edited by N. J. A. Sloane, Dec 19 2008

More terms from R. J. Mathar, Jan 22 2009

A047858 T(n, k) = 2^(k-1)(k + 2n) - n + 1, array read by descending antidiagonals.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 13, 8, 4, 1, 33, 20, 11, 5, 1, 81, 48, 27, 14, 6, 1, 193, 112, 63, 34, 17, 7, 1, 449, 256, 143, 78, 41, 20, 8, 1, 1025, 576, 319, 174, 93, 48, 23, 9, 1, 2305, 1280, 703, 382, 205, 108, 55, 26, 10, 1, 5121, 2816, 1535, 830, 445, 236, 123, 62, 29, 11, 1

Offset: 0

Clark Kimberling

Previous name was: Array T read by diagonals; n-th difference of (T(k,n),T(k,n-1),...,T(k,0)) is k+n, for n=1,2,3,...; k=0,1,2,...

			From _Stefano Spezia_, Jan 03 2023: (Start)
The array begins:
  1, 2,  5, 13,  33,  81,...
  1, 3,  8, 20,  48, 112,...
  1, 4, 11, 27,  63, 143,...
  1, 5, 14, 34,  78, 174,...
  1, 6, 17, 41,  93, 205,...
  1, 7, 20, 48, 108, 236,...
  ...
(End)

Row 1 = (1, 2, 5, 13, 33, ...) = A005183.

Row 2 = (1, 3, 8, 20, 48, ...) = A001792.

T[n_,k_]:=2^(k-1)*(k+2n)-n+1;Table[Reverse[Table[T[n-k,k],{k,0,n}]],{n,0,10}]//Flatten (* Stefano Spezia, Jan 02 2023 *)

T(n, k) = 2^(k-1)*(k + 2*n) - n + 1. - Benoit Cloitre, Jun 17 2003

G.f.: (1 - x - 3*y + 4*x*y + 3*y^2 - 5*x*y^2)/((1 - x)^2*(1 - 2*y)^2*(1 - y)). - Stefano Spezia, Jan 02 2023

New name using formula by Benoit Cloitre, Joerg Arndt, Jan 03 2023

A084858 Binomial transform of A001651.

Original entry on oeis.org

1, 3, 9, 24, 60, 144, 336, 768, 1728, 3840, 8448, 18432, 39936, 86016, 184320, 393216, 835584, 1769472, 3735552, 7864320, 16515072, 34603008, 72351744, 150994944, 314572800, 654311424, 1358954496, 2818572288, 5838471168, 12079595520

Offset: 0

Paul Barry, Jun 11 2003

a(n+1)/3 = A001792(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

I:=[1, 3, 9]; [n le 3 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jun 24 2012

CoefficientList[Series[(x^2-x+1)/(1-2x)^2,{x,0,40}],x] (* Vincenzo Librandi, Jun 24 2012 *)

a(n)=3*(0^n/3+2^n+n<Charles R Greathouse IV, Nov 11 2011

G.f.: (x^2 - x + 1)/(1-2*x)^2.
a(n) = 3*(0^n/3 + 2^n + n*2^n)/4.
For n > 1: a(n) = 2*a(n-1) + 3*2^(n-2). - Philippe Deléham, Nov 10 2011
a(n) = 4*a(n-1) - 4*a(n-2). - Vincenzo Librandi, Jun 24 2012

A007403 a(n) = Sum_{m=0..n} (Sum_{k=0..m} binomial(n,k))^3 = (n+2)2^(3n-1) - 32^(n-2)nbinomial(2n,n).

Original entry on oeis.org

1, 9, 92, 920, 8928, 84448, 782464, 7130880, 64117760, 570166784, 5023524864, 43915595776, 381350330368, 3292451880960, 28283033157632, 241884640182272, 2060565937127424, 17492250190544896, 148027589475696640

Offset: 0

N. J. A. Sloane, Mira Bernstein

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1104 (terms 0..200 from Vincenzo Librandi)
G. E. Andrews and P. Paule, MacMahon's partition analysis. IV. Hypergeometric multisums, In The Andrews Festschrift (Maratea, 1998). Sem. Lothar. Combin. 42 (1999), Art. B42i, 24 pp.
N. J. Calkin, A curious binomial identity, Discr. Math., 131 (194), 335-337.
Bing He, Some identities involving the partial sum of q-binomial coefficients, Electronic J. Combin,, 21 (2014), #P3.17. Gives generalizations. - _N. J. A. Sloane_, Jul 26 2014
M. Hirschhorn, Calkin's binomial identity, Discr. Math., 159 (1996), 273-278.
C. L. Mallows, Letter to N. J. Calkin [Included with permission]
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

If the exponent E in a(n) = Sum_{m=0..n} (Sum_{k=0..m} C(n,k))^E is 1, 2, 3, 4, 5 we get A001792, A003583, A007403, A294435, A294436 respectively.

List([0..20],n->Sum([0..n],m->Sum([0..m],k->Binomial(n,k))^3)); # Muniru A Asiru, Aug 15 2018

[(n+2)*2^(3*n-1)-3*2^(n-2)*n*Binomial(2*n,n): n in [0..20]]; // Vincenzo Librandi, Jul 27 2014

f:=n->n*8^n/2+8^n-(3*n/4)*2^n*binomial(2*n,n);
[seq(f(n),n=0..50)];
A:=proc(n,k) local j; add(binomial(n,j),j=0..k); end;
S:=proc(n,p) local i; global A; add(A(n,i)^p, i=0..n); end;
[seq(S(n,3),n=0..50)]; # N. J. A. Sloane, Nov 17 2017

Table[(n+2)2^(3n-1)-3 2^(n-2)n Binomial[2n,n],{n,0,20}] (* Harvey P. Dale, Jun 30 2011 *)
CoefficientList[Series[(1 - (4 + 3 Sqrt[1 - 8 x]) x)/(1 - 8 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Jul 27 2014 *)
nmax = 18; CoefficientList[Series[Exp[8 x] (1 + 4 x) - 3 x Exp[4 x] (BesselI[0, 4 x] + BesselI[1, 4 x]), {x, 0, nmax}], x] Range[0, nmax]! (* Ilya Gutkovskiy, Aug 18 2018 *)

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

G.f.: (1 - (4 + 3*sqrt(1 - 8*x))*x)/(1 - 8*x)^2. - Harvey P. Dale, Jun 30 2011
E.g.f.: exp(8*x)*(1 + 4*x) - 3*x*exp(4*x)*(BesselI(0,4*x) + BesselI(1,4*x)). - Ilya Gutkovskiy, Aug 15 2018
a(n) ~ n * 2^(3*n-1) * (1 - 3/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Aug 18 2018

A124182 A skewed version of triangular array A081277.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 3, 4, 0, 0, 1, 8, 8, 0, 0, 0, 5, 20, 16, 0, 0, 0, 1, 18, 48, 32, 0, 0, 0, 0, 7, 56, 112, 64, 0, 0, 0, 0, 1, 32, 160, 256, 128, 0, 0, 0, 0, 0, 9, 120, 432, 576, 256, 0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512

Offset: 0

Philippe Deléham, Dec 05 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938. Falling diagonal sums in A052980.

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 3, 4;
  0, 0, 1, 8,  8;
  0, 0, 0, 5, 20, 16;
  0, 0, 0, 1, 18, 48,  32;
  0, 0, 0, 0,  7, 56, 112,  64;
  0, 0, 0, 0,  1, 32, 160, 256,  128;
  0, 0, 0, 0,  0,  9, 120, 432,  576,  256;
  0, 0, 0, 0,  0,  1,  50, 400, 1120, 1280, 512;

Cf. A025192 (column sums). Diagonals include A011782, A001792, A001793, A001794, A006974, A006975, A006976.

T(0,0)=T(1,1)=1, T(n,k)=0 if n < k or if k < 0, T(n,k) = T(n-2,k-1) + 2*T(n-1,k-1).
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A090965(n), (-1)^n*A084120(n), (-1)^n*A006012(n), A033999(n), A000007(n), A001333(n), A084059(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
Sum_{k=0..floor(n/2)} T(n-k,k) = Fibonacci(n-1) = A000045(n-1).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 respectively. - Philippe Deléham, Dec 26 2007
Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x= 0,1,2,3,4,5,6 respectively. - Philippe Deléham, Nov 14 2008
G.f.: (1-y*x)/(1-2y*x-y*x^2). - Philippe Deléham, Dec 04 2011
Sum_{k=0..n} T(n,k)^2 = A002002(n) for n > 0. - Philippe Deléham, Dec 04 2011

A228366 Toothpick sequence from a diagram of compositions of the positive integers (see Comments lines for definition).

Original entry on oeis.org

0, 2, 6, 8, 15, 17, 21, 23, 35, 37, 41, 43, 50, 52, 56, 58, 79, 81, 85, 87, 94, 96, 100, 102, 114, 116, 120, 122, 129, 131, 135, 137, 175, 177, 181, 183, 190, 192, 196, 198, 210, 212, 216, 218, 225, 227, 231, 233, 254, 256, 260, 262, 269, 271, 275

Offset: 0

Omar E. Pol, Aug 22 2013

nonn

In order to construct this sequence we use the following rules:
We start in the first quadrant of the square grid with no toothpicks, so a(0) = 0.
At stage n we place the smallest possible number of toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, n) such that the x-coordinate of the exposed endpoint of the last toothpick is not equal to the x-coordinate of any outer corner of the structure. Then we place toothpicks of length 1 connected by their endpoints in vertical direction, starting from the exposed toothpick endpoint, downward up to touch the structure or up to touch the x-axis.
The sequence gives the number of toothpicks after n stages. A228367 (the first differences) gives the number of toothpicks added at the n-th stage.
Note that the number of toothpick of added at n-th stage in horizontal direction is also A001511(n) and the number of toothpicks added at n-th stage in vertical direction is also A006519(n). Also counting both the x-axis and the y-axis we have that A001511(n) is also the largest part of the n-th region of the diagram and A006519(n) is also the number of parts of the n-th region of the diagram.
After 2^k stages a new section of the structure is completed, so the structure can be interpreted as a diagram of the 2^(k-1) compositions of k in colexicographic order, if k >= 1 (see A228525). The infinite diagram can be interpreted as a table of compositions of the positive integers.

			Illustration of initial terms (n = 1..8):
.                                                _ _ _ _
.                                        _       _      |
.                                _ _     _|_     _|_    |
.                        _       _  |    _  |    _  |   |
.                _ _ _   _|_ _   _|_|_   _|_|_   _|_|_  |
.          _     _    |  _    |  _    |  _    |  _    | |
.    _ _   _|_   _|_  |  _|_  |  _|_  |  _|_  |  _|_  | |
._   _  |  _  |  _  | |  _  | |  _  | |  _  | |  _  | | |
. |   | |   | |   | | |   | | |   | | |   | | |   | | | |
.
.2    6     8      15      17      21      23       35
.
After 16 stages there are 79 toothpicks in the structure which represents the compositions of 5 in colexicographic order as shown below:
-------------------------------
n     Diagram      Composition
-------------------------------
.     _ _ _ _ _
16    _        |   5
15    _|_      |   1+4
14    _  |     |   2+3
13    _|_|_    |   1+1+3
12    _    |   |   3+2
11    _|_  |   |   1+2+2
10    _  | |   |   2+1+2
9     _|_|_|_  |   1+1+1+2
8     _      | |   4+1
7     _|_    | |   1+3+1
6     _  |   | |   2+2+1
5     _|_|_  | |   1+1+2+1
4     _    | | |   3+1+1
3     _|_  | | |   1+2+1+1
2     _  | | | |   2+1+1+1
1      | | | | |   1+1+1+1+1
.

Cf. A001511, A005187, A006519, A011782, A001792, A065120, A139250, A228367, A228370, A228371, A228525.

def A228366(n): return sum(((m:=(i>>1)+1)&-m).bit_length() if i&1 else (m:=i>>1)&-m for i in range(1,2*n+1)) # Chai Wah Wu, Jul 15 2022

a(n) = sum_{k=1..n} (A001511(k) + A006519(k)), n >= 1.
a(n) = A005187(n) + A065120(n-1), n >= 1.
a(n) = A228370(2n).

cp's OEIS Frontend

A053088 a(n) = 3*a(n-2) + 2*a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

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A047858 T(n, k) = 2^(k-1)*(k + 2*n) - n + 1, array read by descending antidiagonals.

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A084858 Binomial transform of A001651.

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A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

A047858 T(n, k) = 2^(k-1)(k + 2n) - n + 1, array read by descending antidiagonals.

A007403 a(n) = Sum_{m=0..n} (Sum_{k=0..m} binomial(n,k))^3 = (n+2)2^(3n-1) - 32^(n-2)nbinomial(2n,n).