A038207 -id:A038207 - cp's OEIS frontend

A140325 a(n) = binomial(n+8,8) * 2^n.

1, 18, 180, 1320, 7920, 41184, 192192, 823680, 3294720, 12446720, 44808192, 154791936, 515973120, 1666990080, 5239111680, 16066609152, 48199827456, 141764198400, 409541017600, 1163958681600, 3259084308480, 9001280471040

Offset: 0

Zerinvary Lajos, Jun 23 2008

With a different offset, number of n-permutations (n>=8) of 3 objects: u, v, z with repetition allowed, containing exactly eight (8) u's. See example.

Number of 8D hypercubes in an (n+8)-dimensional hypercube. [Zerinvary Lajos, Jan 29 2010]

			Example: a(1)=18 because we have uuuuuuuuv, uuuuuuuvu, uuuuuuvuu, uuuuuvuuu, uuuuvuuuu, uuuvuuuuu, uuvuuuuuu, uvuuuuuuu, vuuuuuuuu, uuuuuuuuz, uuuuuuuzu, uuuuuuzuu, uuuuuzuuu, uuuuzuuuu, uuuzuuuuu, uuzuuuuuu, uzuuuuuuu and zuuuuuuu.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (18,-144,672,-2016,4032,-5376,4608,-2304,512).

Cf. A038207, A054851.

[2^n*Binomial(n+8, 8): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

Maple
```
seq(binomial(n+8,8)*2^n,n=0..28);
```

Table[Binomial[n + 8, 8] 2^n, {n, 0, 20}] (* Zerinvary Lajos, Jan 29 2010 *)

a(n) = A038207(n+8,8).
G.f.: 1/(1-2*x)^9. - R. J. Mathar, Feb 11 2010
a(n) = 2*a(n-1) + A054851(n-1). - Ruskin Harding, May 12 2013
a(n) = Sum_{i=8..n+8} binomial(i,8)*binomial(n+8,i). Example: for n=6, a(6) = 1*3003 + 9*2002 + 45*1001 + 165*364 + 495*91 + 1287*14 + 3003*1 = 192192. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 1276/105 - 16*log(2).
Sum_{n>=0} (-1)^n/a(n) = 34992*log(3/2) - 496548/35. (End)

A027466 Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).

Original entry on oeis.org

1, 7, 1, 49, 14, 1, 343, 147, 21, 1, 2401, 1372, 294, 28, 1, 16807, 12005, 3430, 490, 35, 1, 117649, 100842, 36015, 6860, 735, 42, 1, 823543, 823543, 352947, 84035, 12005, 1029, 49, 1, 5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 1

Offset: 0

Olivier Gérard

T(i,j) is the number of i-permutations of 8 objects a,b,c,d,e,f,g,h, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007

Triangle of coefficients in the expansion of (7 + x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 21 2018

			Triangle begins:
        1;
        7,       1;
       49,      14,       1;
      343,     147,      21,      1;
     2401,    1372,     294,     28,      1;
    16807,   12005,    3430,    490,     35,     1;
   117649,  100842,   36015,   6860,    735,    42,    1;
   823543,  823543,  352947,  84035,  12005,  1029,   49,  1;
  5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Harvey P. Dale, Table of n, a(n) for n = 0..5000
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A007318, A038207.

Cf. A317014

Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*7^(i-j)))); # Muniru A Asiru, Jul 21 2018

for i from 0 to 8 do seq(binomial(i, j)*7^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

Flatten[Table[Binomial[i,j]7^(i-j),{i,0,10},{j,0,i}]] (* Harvey P. Dale, Dec 03 2012 *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 7 t[n - 1, k] + t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Zagros Lalo, Jul 21 2018 *)
Table[CoefficientList[ Expand[(7 + x)^n], x], {n, 0, 8}] // Flatten  (* Zagros Lalo, Jul 22 2018 *)

Cube of lower triangular normalized Binomial matrix.
Numerators of lower triangle of (a( i, j ))^3 where a( i, j ) = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 otherwise.
T(0,0) = 1; T(n,k) = 7*T(n-1,k) + T(n-1,k-1) for k = 0..n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 21 2018

Simpler definition from N. J. A. Sloane

A038255 Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).

Original entry on oeis.org

1, 6, 1, 36, 12, 1, 216, 108, 18, 1, 1296, 864, 216, 24, 1, 7776, 6480, 2160, 360, 30, 1, 46656, 46656, 19440, 4320, 540, 36, 1, 279936, 326592, 163296, 45360, 7560, 756, 42, 1, 1679616, 2239488, 1306368, 435456, 90720, 12096, 1008

Offset: 0

N. J. A. Sloane

T(n,k) = A013613(n,n-k), 0 <= k <= n. - Reinhard Zumkeller, Nov 21 2013

			1
6, 1
36, 12, 1
216, 108, 18, 1
1296, 864, 216, 24, 1
7776, 6480, 2160, 360, 30, 1
46656, 46656, 19440, 4320, 540, 36, 1
279936, 326592, 163296, 45360, 7560, 756, 42, 1
1679616, 2239488, 1306368, 435456, 90720, 12096, 1008, 48, 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A038207.

Cf. A000420 (row sums), A013613 (mirrored), A110440, A007318, A000400.

a038255 n k = a038255_tabl !! n !! k
a038255_row n = a038255_tabl !! n
a038255_tabl = map reverse a013613_tabl
-- Reinhard Zumkeller, Nov 21 2013

for i from 0 to 8 do seq(binomial(i, j)*6^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

Table[Binomial[n,m]6^(n-m),{n,0,10},{m,0,n}]//Flatten (* Harvey P. Dale, Dec 25 2019 *)

G.f.: 1/(1 - 6*x - x*y). - Ilya Gutkovskiy, Apr 21 2017

A083102 a(n) = 2a(n-1) + 10a(n-2), with a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 14, 48, 236, 952, 4264, 18048, 78736, 337952, 1463264, 6306048, 27244736, 117549952, 507547264, 2190594048, 9456660736, 40819261952, 176205131264, 760602882048, 3283257076736, 14172542973952, 61177656715264, 264080743170048, 1139938053492736, 4920683538685952

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

a(n+1) = a(n) + A083101(n). A083101(n)/a(n) converges to sqrt(11).
Antidiagonals of A038207. - Mark Dols, Aug 31 2009
Numerators of stationary probabilities for M2/M/1 queue system. In this queue, customers arrive in groups of 2. Intensity of arrival = 2. Service rate = 5. There is only one server and an infinite queue. - Igor Kleiner, Nov 02 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (2,10).

Cf. A038207, A083101.

I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1) + 10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

CoefficientList[Series[1/(1-2x-10x^2), {x, 0, 25}], x]
LinearRecurrence[{2,10}, {1,2}, 30] (* G. C. Greubel, Jan 08 2018 *)

x='x+O('x^30); Vec(1/(1-2*x-10*x^2)) \\ G. C. Greubel, Jan 08 2018

[lucas_number1(n,2,-10) for n in range(1, 24)] # Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1-2*x-10*x^2).
From Paul Barry, Sep 29 2004: (Start)
E.g.f.: exp(x) * sinh(sqrt(11)*x) / sqrt(11).
a(n) = Sum_{k=0..n} binomial(n,2*k+1) * 11^k. (End)
a(n) = ((1+sqrt(11))^n - (1-sqrt(11))^n)/(2*sqrt(11)). - Rolf Pleisch, Jul 06 2009
G.f.: G(0)/(2-2*x), where G(k)= 1 + 1/(1 - x*(11*k-1)/( x*(11*k+10) - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 14 2013
G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+2 + 10*x )/( x*(4*k+4 + 10*x ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013

A140354 a(n) = binomial(n+9,9)*2^n.

Original entry on oeis.org

1, 20, 220, 1760, 11440, 64064, 320320, 1464320, 6223360, 24893440, 94595072, 343982080, 1203937280, 4074864640, 13388840960, 42844291072, 133888409600, 409541017600, 1228623052800, 3621204787200, 10501493882880, 30004268236800

Offset: 0

Zerinvary Lajos, Jun 23 2008

Number of 9D hypercubes in an (n+9)-dimensional hypercube. - Zerinvary Lajos, Jan 29 2010; corrected by Michel Marcus, Jan 10 2015

			For n=6, a(6) = 1*5005 + 10*3003 + 55*1365 + 220*455 + 715*105 + 2002*15 + 5005*1 = 320320.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (20,-180,960,-3360,8064,-13440,15360,-11520,5120,-1024).

Cf. A038207, A140325.

[2^n*Binomial(n+9, 9): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

Maple
```
seq(binomial(n+9,9)*2^n,n=0..23);
```

Table[Binomial[n + 9, 9] 2^n, {n, 0, 20}] (* Zerinvary Lajos, Jan 29 2010 *)
CoefficientList[Series[1/(1-2x)^10,{x,0,30}],x] (* Harvey P. Dale, Jul 18 2011 *)

a(n)=binomial(n+9,9)<Charles R Greathouse IV, Jul 18 2011

[lucas_number2(n, 2, 0)*binomial(n,9)/512 for n in range(9, 31)] # Zerinvary Lajos, Mar 10 2009

a(n) = A038207(n+9,9).
G.f.: 1/(1-2*x)^10. - Harvey P. Dale, Jul 18 2011
a(0)=1, a(1)=20, a(2)=220, a(3)=1760, a(4)=11440, a(5)=64064, a(6)=320320, a(7)=1464320, a(8)=6223360, a(9)=24893440; for n>9, a(n) = 20*a(n-1) - 180*a(n-2) + 960*a(n-3) - 3360*a(n-4) + 8064*a(n-5) - 13440*a(n-6) + 15360*a(n-7) - 11520*a(n-8) + 5120*a(n-9) - 1024*a(n-10). - Harvey P. Dale, Jul 18 2011
a(n) = 2*a(n-1) + A140325(n-1). - Ruskin Harding, May 13 2013
a(n) = Sum_{i=9..n+9} binomial(i,9)*binomial(n+9,i). - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 18*log(2) - 1599/140.
Sum_{n>=0} (-1)^n/a(n) = 118098*log(3/2) - 6703713/140. (End)

A112857 Triangle T(n,k) read by rows: number of Green's R-classes in the semigroup of order-preserving partial transformations (of an n-element chain) consisting of elements of height k (height(alpha) = |Im(alpha)|).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 15, 17, 7, 1, 1, 31, 49, 31, 9, 1, 1, 63, 129, 111, 49, 11, 1, 1, 127, 321, 351, 209, 71, 13, 1, 1, 255, 769, 1023, 769, 351, 97, 15, 1, 1, 511, 1793, 2815, 2561, 1471, 545, 127, 17, 1, 1, 1023, 4097, 7423, 7937, 5503, 2561, 799, 161, 19, 1

Offset: 0

Abdullahi Umar, Aug 25 2008

Sum of rows of T(n, k) is A007051; T(n,k) = |A118801(n,k)|.
Row-reversed variant of A119258. - R. J. Mathar, Jun 20 2011
Pairwise sums of row terms starting from the right yields triangle A038207. - Gary W. Adamson, Feb 06 2012
Riordan array (1/(1 - x), x/(1 - 2*x)). - Philippe Deléham, Jan 17 2014
Appears to coincide with the triangle T(n,m) (n >= 1, 1 <= m <= n) giving number of set partitions of [n], avoiding 1232, with m blocks [Crane, 2015]. See also A250118, A250119. - N. J. A. Sloane, Nov 25 2014
(A007318)^2 = A038207 = T*|A167374|. See A118801 for other relations to the Pascal matrix. - Tom Copeland, Nov 17 2016

			T(3,2) = 5 because in a regular semigroup of transformations the Green's R-classes coincide with convex partitions of subsets of {1,2,3} with convex classes (modulo the subsets): {1}, {2}/{1}, {3}/{2}, {3}/{1,2}, {3}/{1}, {2,3}
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,    1;
  1,    3,    1;
  1,    7,    5,    1;
  1,   15,   17,    7,    1;
  1,   31,   49,   31,    9,    1;
  1,   63,  129,  111,   49,   11,    1;
  1,  127,  321,  351,  209,   71,   13,   1;
  1,  255,  769, 1023,  769,  351,   97,  15,   1;
  1,  511, 1793, 2815, 2561, 1471,  545, 127,  17,  1;
  1, 1023, 4097, 7423, 7937, 5503, 2561, 799, 161, 19, 1;
  ...
As to matrix M, top row of M^3 = (1, 7, 5, 1, 0, 0, 0, ...)

Peter Bala, A 4-parameter family of embedded Riordan arrays
Peter Bala, A note on the diagonals of a proper Riordan Array
Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.

Cf. A007051, A118801, A135233, columns: A000225, A000337, A055580, A027608.
Cf. also A038207, A250118, A250119.
Cf. A007318, A167374, A145661, A029635.

A112857 := proc(n,k) if k=0 or k=n then 1; elif k <0 or k>n then 0; else 2*procname(n-1,k)+procname(n-1,k-1) ; end if; end proc: # R. J. Mathar, Jun 20 2011

Table[Abs[1 + (-1)^k*2^(n - k + 1)*Sum[ Binomial[n - 2 j - 2, k - 2 j - 1], {j, 0, Floor[k/2]}]] - 4 Boole[And[n == 1, k == 0]], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

T(n,k) = Sum_{j = k..n} C(n,j)*C(j-1,k-1).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) for n >= 2 and 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.
n-th row = top row of M^n, deleting the zeros, where M is an infinite square production matrix with (1,1,1,...) as the superdiagonal and (1,2,2,2,...) as the main diagonal. - Gary W. Adamson, Feb 06 2012
From Peter Bala, Mar 05 2018 (Start):
The following remarks are particular cases of more general results for Riordan arrays of the form (f(x), x/(1 - k*x)).
Let R(n,x) denote the n-th row polynomial of this triangle. The polynomial R(n,2*x) has the e.g.f. Sum_{k = 0..n} T(n,k)*(2*x)^k/k!. The e.g.f. for the n-th diagonal of the triangle (starting at n = 0 for the main diagonal) equals exp(x) * the e.g.f. for the polynomial R(n,2*x). For example, when n = 3 we have exp(x)*(1 + 7*(2*x) + 5*(2*x)^2/2! + (2*x)^3/3!) = 1 + 15*x + 49*x^2/2! + 111*x^3/3! + 209*x^4/4! + ....
Let P(n,x) = Sum_{k = 0..n} T(n,k)*x^(n-k) denote the n-th row polynomial in descending powers of x. Then P(n,x) is the n-th degree Taylor polynomial of the function (1 + 2*x)^n/(1 + x) about 0. For example, for n = 4 we have (1 + 2*x)^4/(1 + x) = x^4 + 15*x^3 + 17*x^2 + 7*x + 1 + O(x^5).
See A118801 for a signed version of this triangle and A145661 for a signed version of the row reversed triangle. (End)
Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 - 2*x)/((1 - x)*(1 - 2*x - x*y)). - Petros Hadjicostas, Feb 14 2021
The matrix inverse of the Lucas triangle A029635 is -T(n, k)/(-2)^(n-k+1). - Peter Luschny, Dec 22 2024

A041181 Denominators of continued fraction convergents to sqrt(101).

Original entry on oeis.org

1, 20, 401, 8040, 161201, 3232060, 64802401, 1299280080, 26050404001, 522307360100, 10472197606001, 209966259480120, 4209797387208401, 84405914003648140, 1692328077460171201, 33930967463207072160, 680311677341601614401

Offset: 0

N. J. A. Sloane

Generalized Pell numbers (A000129). Antidiagonals of A038207. - Mark Dols, Aug 31 2009
a(n) equals the number of words of length n on alphabet {0,1,...,20} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, May 03 2023: (Start)
Also called the 20-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 20 kinds of squares available. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (20,1).

Cf. A000129, A038207, A040090, A041180.
Cf. similar sequences listed in A243399.
Row n=20 of A073133, A172236 and A352361 and column k=20 of A157103.

[n le 2 select (20)^(n-1) else 20*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013

Denominator[Convergents[Sqrt[101], 30]] (* Vincenzo Librandi, Dec 12 2013 *)
LinearRecurrence[{20,1},{1,20},20] (* Harvey P. Dale, Mar 17 2020 *)

A041181=BinaryRecurrenceSequence(20,1,1,20)
[A041181(n) for n in range(31)] # G. C. Greubel, Sep 29 2024

a(n) = Fibonacci(n+1, 20), the n-th Fibonacci polynomial evaluated at x=20. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 20*a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=20.
G.f.: 1/(1-20*x-x^2). (End)

A065109 Triangle T(n,k) of coefficients relating to Bezier curve continuity.

Original entry on oeis.org

1, 2, -1, 4, -4, 1, 8, -12, 6, -1, 16, -32, 24, -8, 1, 32, -80, 80, -40, 10, -1, 64, -192, 240, -160, 60, -12, 1, 128, -448, 672, -560, 280, -84, 14, -1, 256, -1024, 1792, -1792, 1120, -448, 112, -16, 1, 512, -2304, 4608, -5376, 4032, -2016, 672, -144, 18, -1, 1024, -5120, 11520, -15360, 13440

Offset: 0

Peter J. Taylor, Nov 12 2001

Row sums are 1, antidiagonal sums are the natural numbers. - Gerald McGarvey, May 29 2005
Row sums = 1. - Roger L. Bagula, Sep 12 2008
Riordan array (1/(1-2x), -x/(1-2x)). - Philippe Deléham, Nov 27 2009
Triangle T(n,k), read by rows, given by [2,0,0,0,0,0,0,0,...] DELTA [ -1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2009

			For C-2 continuity between P and Q we require Q_0 = P_n; Q_1 = 2P_n - P_n-1; Q_2 = 4P_n - 4P_n-1 + P_n-2.
Triangle begins:
     1;
     2,     -1;
     4,     -4,     1;
     8,    -12,     6,     -1;
    16,    -32,    24,     -8,     1;
    32,    -80,    80,    -40,    10,     -1;
    64,   -192,   240,   -160,    60,    -12,     1;
   128,   -448,   672,   -560,   280,    -84,    14,    -1;
   256,  -1024,  1792,  -1792,  1120,   -448,   112,   -16,    1;
   512,  -2304,  4608,  -5376,  4032,  -2016,   672,  -144,   18,   -1;
  1024,  -5120, 11520, -15360, 13440,  -8064,  3360,  -960,  180,  -20,  1;
  2048, -11264, 28160, -42240, 42240, -29568, 14784, -5280, 1320, -220, 22, -1;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
Peter J. Taylor, Conditions for C-a Continuity of Bezier Curves

Cf. A038207, A013609. Apart from signs, same as A038207.

a065109 n k = a065109_tabl !! n !! k
a065109_row n = a065109_tabl !! n
a065109_tabl = iterate
   (\row -> zipWith (-) (map (* 2) row ++ [0]) ([0] ++ row)) [1]
-- Reinhard Zumkeller, Apr 25 2013

/* As triangle: */  [[(-1)^k*2^(n-k)*Binomial(n, k): k in [0..n]]: n in [0..15]]; // Vincenzo Librandi, Apr 26 2015

seq(seq((-1)^k * 2^(n-k) * binomial(n, k), k= 0 .. n), n = 0 .. 12); # Robert Israel, Apr 26 2015

t[n_, m_, k_] = (-1)^m*Multinomial[n - m - k, m, k]; Table[Table[Sum[t[n, m, k], {k, 0, n}], {m, 0, n}], {n, 0, 11}]; Flatten[%] (* Roger L. Bagula, Sep 12 2008 *)
Flatten[Table[(-1)^k 2^(n-k) Binomial[n,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Mar 13 2013 *)

T(n, k) = (-1)^k * 2^(n-k) * binomial(n, k).
Sum_{i=0..n} binomial(n,i) * (-1)^i * T(i,r) = (-1)^(n-r) * binomial(n,r).
For n > 0, T(n, k) = 2*T(n-1, k) - T(n-1, k-1). - Gerald McGarvey, May 29 2005
p(n,m,k) = (-1)^m*multinomial(n - m - k, m, k); t(n,m) = Sum_{k=0..n} (-1)^m*multinomial(n - m - k, m, k). - Roger L. Bagula, Sep 12 2008
Sum_{k=0..n} T(n,k)*A000108(k) = A001405(n). - Philippe Deléham, Nov 27 2009
Sum_{k=0..n} T(n,k)*x^k = (2-x)^n. - Philippe Deléham, Dec 15 2009
G.f.: Sum_{n>=0} (2-x)^n * x^(n*(n+1)/2). - Robert Israel, Apr 26 2015
G.f.: 1/(1-2*x+x*y). - R. J. Mathar, Aug 11 2015

A080928 Triangle T(n,k) read by rows: T(n,k) = Sum_{i=0..n} C(n,2i)*C(2i,k).

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 4, 6, 3, 0, 8, 16, 12, 4, 1, 16, 40, 40, 20, 5, 0, 32, 96, 120, 80, 30, 6, 1, 64, 224, 336, 280, 140, 42, 7, 0, 128, 512, 896, 896, 560, 224, 56, 8, 1, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 0, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10

Offset: 0

Paul Barry, Feb 26 2003

Gives the general solution to a(n) = 2*a(n-1) + k(k+2)*a(n-2), a(0) = a(1) = 1. The value k=1 gives the row sums of the triangle, or 1,1,5,13,... This is A046717, the solution to a(n) = 2*a(n-1) + 3*a(n-2), a(0)=a(1)=1.

Product of A007318 and A007318 with every odd-indexed row set to zero. - Paul Barry, Nov 08 2005

			Triangle begins:
    1;
    1,    0;
    2,    2,    1;
    4,    6,    3,    0;
    8,   16,   12,    4,    1;
   16,   40,   40,   20,    5,    0;
   32,   96,  120,   80,   30,    6,   1;
   64,  224,  336,  280,  140,   42,   7,  0;
  128,  512,  896,  896,  560,  224,  56,  8, 1;
  256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 0; etc.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 156.
J-L. Kim, Relation between weight distribution and combinatorial identities, Bulletin of the Institute of Combinatorics and its Applications, Canada, 31, 2001, pp. 69-79.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150).
Saul Schleimer, Bert Wiest, On the conjugacy problem in braid groups: Garside theory and subsurfaces, arXiv:1807.01500 [math.GT], 2018.

Apart from k=n, T(n, k) equals (1/2)*A038207(n, k).

Columns include A011782, 2*A001792, A080929, 4*A080930. Row sums are in A046717.

Table[Sum[Binomial[n, 2 i] Binomial[2 i, k], {i, 0, n}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 11 2018 *)

T(n, n) = (n+1) mod 2, T(n, k) = C(n, k)*2^(n-k-1).
T(n, 0) = A011782(n), T(n, k)=0, k>n, T(2n, 2n)=1, T(2n-1, 2n-1)=0, T(n+1, n)=n+1. Otherwise T(n, k) = T(n-1, k-1) + 2T(n-1, k). Rows are the coefficients of the polynomials in the expansion of (1-x)/((1+k*x)*(1-(k+2)*x)). The main diagonal is 1, 0, 1, 0, 1, 0, ... with g.f. 1/(1-x^2). Subsequent subdiagonals are given by A011782(k)*C(n+k, k) with g.f. A011782(k)/(1-x)^k.
T(n, k) = Sum_{j=0..n} C(n, j)*C(j, k)*(1+(-1)^j)/2; T(n, k) = 2^(n-k-1)*(C(n, k) + (-1)^n*C(0, n-k)). - Paul Barry, Nov 08 2005

Edited by Ralf Stephan, Feb 04 2005

A099089 Riordan array (1, 2+x).

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 0, 4, 8, 0, 0, 1, 12, 16, 0, 0, 0, 6, 32, 32, 0, 0, 0, 1, 24, 80, 64, 0, 0, 0, 0, 8, 80, 192, 128, 0, 0, 0, 0, 1, 40, 240, 448, 256, 0, 0, 0, 0, 0, 10, 160, 672, 1024, 512, 0, 0, 0, 0, 0, 1, 60, 560, 1792, 2304, 1024, 0, 0, 0, 0, 0, 0, 12, 280, 1792, 4608, 5120, 2048

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A000129. Diagonal sums are A008346. The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1/2, -1/2, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008
As an upper right triangle (in the example), table rows give number of points, edges, faces, cubes, 4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley, Apr 14 2000. More precisely, the (i,j)-th entry is the number of j-dimensional subspaces of an i-dimensional hypercube (see the Coxeter reference). - Christof Weber, May 08 2009

			Triangle begins:
  1;
  0,  2;
  0,  1,  4;
  0,  0,  4,  8;
  0,  0,  1, 12, 16;
  0,  0,  0,  6, 32, 32;
  0,  0,  0,  1, 24, 80, 64;
The entries can also be interpreted as the antidiagonal reading of the following array:
  1,    2,    4,    8,   16,   32,   64,  128,  256,  512, 1024,... A000079
  0,    1,    4,   12,   32,   80,  192,  448, 1024, 2304, 5120,... A001787
  0,    0,    1,    6,   24,   80,  240,  672, 1792, 4608,11520,... A001788
  0,    0,    0,    1,    8,   40,  160,  560, 1792, 5376,15360,... A001789
  0,    0,    0,    0,    1,   10,   60,  280, 1120, 4032,13440,...
  0,    0,    0,    0,    0,    1,   12,   84,  448, 2016, 8064,...
  0,    0,    0,    0,    0,    0,    1,   14,  112,  672, 3360,...
  0,    0,    0,    0,    0,    0,    0,    1,   16,  144,  960,...
  0,    0,    0,    0,    0,    0,    0,    0,    1,   18,  180,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    1,   20,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    0,    1,...

H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.

Eric W. Weisstein's Mathworld, Hypercube.

Cf. A053118, A008312, A062715, A038207.

Number triangle T(n,k) = binomial(k, n-k)*2^k*(1/2)^(n-k); columns have g.f. (2*x+x^2)^k.
G.f.: 1/(1-2y*x-y*x^2). - Philippe Deléham, Nov 20 2011
Sum_ {k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A090017(n+1), A090018(n), A190510(n+1), A190955(n+1) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Nov 20 2011
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,1) = 1, T(2,2) = 4, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013

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