A013609 -id:A013609 - cp's OEIS frontend

A167591 A triangle related to the a(n) formulas of the rows of the ED4 array A167584.

1, 4, -2, 12, -8, 9, 32, -16, 120, -60, 80, 0, 952, -768, 525, 192, 160, 5664, -5008, 12396, -5670, 448, 896, 27888, -20672, 162740, -133128, 72765, 1024, 3584, 120064, -46720, 1537216, -1562464, 2557296, -1081080, 2304, 12288, 467712, 76800

Offset: 1

Johannes W. Meijer, Nov 10 2009

The a(n) formulas given below correspond to the first ten rows of the ED4 array A167584.

The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

			Row 1: a(n) = 1.
Row 2: a(n) = 4*n - 2.
Row 3: a(n) = 12*n^2 - 8*n + 9.
Row 4: a(n) = 32*n^3 - 16*n^2 + 120*n - 60.
Row 5: a(n) = 80*n^4 + 0*n^3 + 952*n^2 - 768*n + 525.
Row 6: a(n) = 192*n^5 + 160*n^4 + 5664*n^3 - 5008*n^2 + 12396*n - 5670.
Row 7: a(n) = 448*n^6 + 896*n^5 + 27888*n^4 - 20672*n^3 + 162740*n^2 - 133128*n + 72765.
Row 8: a(n) = 1024*n^7 + 3584*n^6 + 120064*n^5 - 46720*n^4 + 1537216*n^3 - 1562464*n^2 + 2557296*n - 1081080.
Row 9: a(n) = 2304*n^8 + 12288*n^7 + 467712*n^6 + 76800*n^5 + 11589216*n^4 - 12058368*n^3 + 47963568*n^2 - 38278080*n + 18243225.
Row 10: a(n) = 5120*n^9 + 38400*n^8 + 1686528*n^7 + 1540608*n^6 + 73898880*n^5 - 66179520*n^4 + 631348672*n^3 - 669559008*n^2 + 869709780*n - 344594250.

A167584 is the ED4 array.
A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the ED4 array.
A001787, A167592, A167593, A168307 and A168308 equal the first five left hand triangle columns.
A001193 equals the first right hand triangle column.
A024199 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Comment and formulas added by Johannes W. Meijer, Nov 23 2009

A126443 a(n) = Sum_{k=0..n-1} C(n-1,k)a(k)2^k for n>0, with a(0)=1.

Original entry on oeis.org

1, 1, 3, 17, 179, 3489, 127459, 8873137, 1195313043, 315321098561, 164239990789571, 169810102632595281, 349630019758589841523, 1436268949679165936016097, 11784559509424676876673518499, 193243076262167105764611875139569

Offset: 0

Paul D. Hanna, Jan 01 2007

nonn

Generated by a generalization of a recurrence for the Bell numbers (A000110).

Starting with offset 1 = eigensequence of triangle A013609. - Gary W. Adamson, Sep 04 2009

Seiichi Manyama, Table of n, a(n) for n = 0..81

Cf. A126347, A000110.
Cf. A013609. - Gary W. Adamson, Sep 04 2009
Column k=2 of A306245.

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

a(n) = Sum_{k=0..n*(n-1)/2} A126347(n,k)*2^k.
G.f. A(x) satisfies: A(x) = 1 + x*A(2*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Sep 02 2019
a(n) ~ c * 2^(n*(n-1)/2), where c = A081845 = 4.7684620580627434482997985... - Vaclav Kotesovec, Sep 16 2019

A167580 A triangle related to the a(n) formulas of the rows of the ED3 array A167572.

Original entry on oeis.org

1, 6, -1, 20, 0, 3, 56, 28, 98, -15, 144, 192, 1080, -48, 105, 352, 880, 7568, 2024, 6534, -945, 832, 3328, 40976, 31616, 132444, -8112, 10395, 1920, 11200, 187488, 274480, 1593960, 286900, 972162, -135135, 4352, 34816, 761600, 1784320, 13962848

Offset: 1

Johannes W. Meijer, Nov 10 2009

The a(n) formulas given below correspond to the first ten rows of the ED3 array A167572.

The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

			Row 1: a(n) = 1.
Row 2: a(n) = 6*n - 1.
Row 3: a(n) = 20*n^2 + 0*n + 3.
Row 4: a(n) = 56*n^3 + 28*n^2 + 98*n - 15.
Row 5: a(n) = 144*n^4 + 192*n^3 + 1080*n^2 - 48*n + 105.
Row 6: a(n) = 352*n^5 + 880*n^4 + 7568*n^3 + 2024*n^2 + 6534*n - 945.
Row 7: a(n) = 832*n^6 + 3328*n^5 + 40976*n^4 + 31616*n^3 + 132444*n^2 - 8112*n + 10395.
Row 8: a(n) = 1920*n^7 + 11200*n^6 + 187488*n^5 + 274480*n^4 + 1593960*n^3 + 286900*n^2 + 972162*n - 135135.
Row 9: a(n) = 4352*n^8 + 34816*n^7 + 761600*n^6 + 1784320*n^5 + 13962848*n^4 + 7874944*n^3 + 29641200*n^2 - 2080800*n + 2027025.
Row 10: a(n) = 9728*n^9 + 102144*n^8 + 2830848*n^7 + 9645312*n^6 + 98382912*n^5 + 106720416*n^4 + 522283552*n^3 + 69265488*n^2 + 255468870*n - 34459425.

A167572 is the ED3 array.
A000012, A016969, A167573, A167574 and A167575 equal the first five rows of the ED3 array.
A014480, A167581, A167582, A168305 and A168306 equal the first five left hand triangle columns.
A001147 equals the first right hand triangle column.
A167576 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Comment and links added by Johannes W. Meijer, Nov 23 2009

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

A001848 Crystal ball sequence for 6-dimensional cubic lattice.

Original entry on oeis.org

1, 13, 85, 377, 1289, 3653, 8989, 19825, 40081, 75517, 134245, 227305, 369305, 579125, 880685, 1303777, 1884961, 2668525, 3707509, 5064793, 6814249, 9041957, 11847485, 15345233, 19665841, 24957661, 31388293, 39146185, 48442297, 59511829, 72616013, 88043969

Offset: 0

N. J. A. Sloane

Number of nodes of degree 12 in virtual, optimal chordal graphs of diameter d(G)=n. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002
Equals binomial transform of [1, 12, 60, 160, 240, 192, 64, 0, 0, 0, ...] where (1, 12, 60, 160, 240, 192, 64) = row 6 of the Chebyshev triangle A013609. - Gary W. Adamson, Jul 19 2008
a(n) is the number of points in Z^6 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array (A008288), as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 6 from any given point. - Shel Kaphan, Jan 02 2023

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 231.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, A local Riemann hypothesis, I pages 16 and 17
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001847, A013609.
Cf. A240876.
Row/column 6 of A008288.

for n from 1 to k do eval(4/45*n^6+4/15*n^5+14/9*n^4+8/3*n^3+196/45*n^2+46/15*n+1); od;
A001848:=-(z+1)**6/(z-1)**7; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[-(z + 1)^6/(z - 1)^7, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)

G.f.: (1+x)^6 /(1-x)^7.
a(n) = (4/45)*n^6 + (4/15)*n^5 + (14/9)*n^4 + (8/3)*n^3 + (196/45)*n^2 + (46/15)*n + 1. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002
a(n) = Sum_{k=0..min(6,n)} 2^k * binomial(6,k)* binomial(n,k). See Bump et al. - Tom Copeland, Sep 05 2014
Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = log(2) - 37/60 = log(2) - (1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6). - Peter Bala, Mar 23 2024

A130123 Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).

Original entry on oeis.org

1, 0, 2, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4096

Offset: 0

Gary W. Adamson, May 11 2007

A 2^n transform matrix.
A130123 * A007318 = A038208. A007318 * A130123 = A013609. A130124 = A130123 * A002260. A130125 = A128174 * A130123.
Triangle T(n,k), 0 <= k <= n, given by [0,0,0,0,0,0,...] DELTA [2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 26 2007
Also the Bell transform of A000038. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
T is the convolution triangle of the characteristic function of 2 (see A357368). - Peter Luschny, Oct 19 2022

			First few terms of the triangle:
  1;
  0, 2;
  0, 0, 4;
  0, 0, 0, 8;
  0, 0, 0, 0, 16;
  0, 0, 0, 0,  0, 32; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A130124, A130125.

[[k eq n select 2^n else 0: k in [0..n]]: n in [0..14]]; // G. C. Greubel, Jun 05 2019

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n=0,2,0), 9); # Peter Luschny, Jan 27 2016
# Uses function PMatrix from A357368.
PMatrix(10, n -> ifelse(n=1, 2, 0)); # Peter Luschny, Oct 19 2022

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 2, 0]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
Table[If[k==n, 2^n, 0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

{T(n,k) = if(k==n, 2^n, 0)}; \\ G. C. Greubel, Jun 05 2019

def T(n, k):
    if (k==n): return 2^n
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..14)] # G. C. Greubel, Jun 05 2019

G.f.: 1/(1-2*x*y). - R. J. Mathar, Aug 11 2015

A105728 Triangle read by rows: T(n,1) = 1, T(n,n) = n and for 1 < k < n: T(n,k) = T(n-1,k-1) + 2*T(n-1,k).

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 11, 11, 4, 1, 23, 33, 19, 5, 1, 47, 89, 71, 29, 6, 1, 95, 225, 231, 129, 41, 7, 1, 191, 545, 687, 489, 211, 55, 8, 1, 383, 1281, 1919, 1665, 911, 321, 71, 9, 1, 767, 2945, 5119, 5249, 3487, 1553, 463, 89, 10, 1, 1535, 6657, 13183, 15617, 12223, 6593, 2479, 641, 109, 11

Offset: 1

Reinhard Zumkeller, Apr 18 2005

Sum of n-th row = 3^(n-1): Sum_{k=1..n} T(n,k) = A000244(n-1);

for n>1: T(n,2) = A083329(n-1), T(n,n-1) = A028387(n-2).

			Triangle begins as:
  1;
  1,  2;
  1,  5,  3;
  1, 11, 11,  4;
  1, 23, 33, 19,  5;
  1, 47, 89, 71, 29, 6;
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A013609, A115068.

a105728 n k = a105728_tabl !! (n-1) !! (k-1)
a105728_row n = a105728_tabl !! (n-1)
a105728_tabl = iterate (\row -> zipWith (+) ([0] ++ tail row ++ [1]) $
                                zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Jul 22 2013

function T(n,k)
  if k eq 1 then return 1;
  elif k eq n then return n;
  else return T(n-1,k-1) + 2*T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 13 2019

T:= proc(n, k) option remember;
      if k=1 then 1
    elif k=n then n
    else T(n-1, k-1) + 2*T(n-1, k)
      fi
    end:
seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Nov 13 2019

T[n_, k_]:= T[n, k]= If[k==1, 1, If[k==n, n, T[n-1, k-1] + 2*T[n-1, k]]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Nov 13 2019 *)

@CachedFunction
def T(n, k):
    if (k==1): return 1
    elif (k==n): return n
    else: return T(n-1,k-1) + 2*T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 13 2019

A115068 Triangle read by rows: T(n,k) = number of elements in the Coxeter group D_n with descent set contained in {s_k}, for 0<=k<=n-1.

Original entry on oeis.org

1, 2, 2, 4, 6, 3, 8, 16, 12, 4, 16, 40, 40, 20, 5, 32, 96, 120, 80, 30, 6, 64, 224, 336, 280, 140, 42, 7, 128, 512, 896, 896, 560, 224, 56, 8, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10, 1024, 5632, 14080, 21120

Offset: 1

Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006

A115068 is the fission of the polynomial sequence (p(x,n)) by the polynomial sequence ((2x+1)^n), where p(n,x)=x^n+x^(n-1)+...+x+1, n>=0. See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011

			First six rows:
1
2...2
4...6....3
8...16...12...4
16..40...40...20...5
32..96...120..80...30...6

A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, New York, 2005.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990.

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A007318, A038207, A193862.

Cf. A105728, A013609.

a115068 n k = a115068_tabl !! (n-1) !! (k-1)
a115068_row n = a115068_tabl !! (n-1)
a115068_tabl = iterate (\row -> zipWith (+) (row ++ [1]) $
                                zipWith (+) (row ++ [0]) ([0] ++ row)) [1]
-- Reinhard Zumkeller, Jul 22 2013

z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + 1;
q[n_, x_] := (2 x + 1)^n;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A115068 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193862 *)

T(n,k)=binomial(n,k)*2^(n-k-1).

T(n,1) = 2^(n-1), T(n,n) = n, for n > 1: T(n,k) = T(n-1,k-1) + 2*T(n-1,k), 1 < k < n. - Reinhard Zumkeller, Jul 22 2013

A006588 a(n) = 4^n(3n)!/((2n)!n!).

Original entry on oeis.org

1, 12, 240, 5376, 126720, 3075072, 76038144, 1905131520, 48199827456, 1228623052800, 31504481648640, 811751838842880, 20999667135283200, 545086744471535616, 14189559697354260480, 370298578584748425216, 9684502341534993088512, 253765034617761850982400

Offset: 0

N. J. A. Sloane

Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Natick, MA, 2004. See p. 26.
William Allen Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 35.
Henry W. Gould, Combinatorial Identities, Morgantown, 1972; The right-hand side of a binomial coefficient identity, Eq. 3.115, page 35.

Vincenzo Librandi, Table of n, a(n) for n = 0..710
Necdet Batir, On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; arXiv preprint, arXiv:math/0512310 [math.AC], 2005.
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 31, eq. (37).
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991.

Cf. A000302, A005809, A006587, A013609.

[4^n*Factorial(3*n)/(Factorial(2*n)* Factorial(n)): n in [0..20]]; // Vincenzo Librandi, Oct 01 2018

A006588 := n->add( binomial(4*n+1,2*n-2*k)*binomial(n+k,k),k=0..n): seq(A006588(n), n=0..15);
h := proc(x) hypergeom([1/3, 2/3], [1/2], 27*x) end: ser := series(h(x), x, 20): seq(coeff(ser, x, n), n=0..15); # Peter Luschny, Sep 30 2018

Table[4^n*(3*n)!/((2*n)!*n!), {n, 0, 20}] (* Erich Friedman, Mar 22 2008 *)

a(n) = 4^n*(3*n)!/((2*n)!*n!) \\ P L Patodia (pannalal(AT)usa.net), Feb 24 2007

a(n) = sum(k=0,n,binomial(4*n+1,2*n-2*k)*binomial(n+k,k)) \\ P L Patodia (pannalal(AT)usa.net), Feb 24 2007

def A006588(n): return 4**n*binomial(3*n,n)
print([A006588(n) for n in range(41)]) # G. C. Greubel, Aug 27 2025

a(n) = Sum_{k=0..n} C(4*n+1, 2*n-2*k)*C(n+k, k) = 4^n*C(3*n, n).
a(n) ~ (1/2)*3^(1/2)*Pi^(-1/2)*n^(-1/2)*3^(3*n)*{1 - (7/72)*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Jun 11 2002
a(n) = A013609(3*n, 2*n). - Johannes W. Meijer, Aug 22 2013
a(n) = [x^n] hypergeom([1/3, 2/3], [1/2], 27*x). - Peter Luschny, Sep 30 2018
a(n) = Sum_{k = n..3*n} binomial(3*n,k)*binomial(k,n). - Peter Bala, Mar 25 2023
From Amiram Eldar, Dec 07 2024: (Start)
a(n) = A000302(n) * A005809(n).
Sum_{n>=1} (-1)^n/a(n) = -1/28 - 3*log(2)/32 + (13/(112*sqrt(7))) * arctan(sqrt(7)/5) (Borwein et al., 2004; Borwein and Girgensohn, 2005; Batir, 2005). (End)
From G. C. Greubel, Aug 27 2025: (Start)
G.f.: (1/(2*(1-27*x))*( cos(t) + cos(2*t) ), where t = (1/3)*arccos(1-54*x).
E.g.f.: hypergeometric2F2([1/3, 2/3], [1/2, 1], 27*x). (End)

A303872 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-1,k-1) for k = 0,1,...,n; T(n,k)=0 for n or k < 0.

Original entry on oeis.org

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

Offset: 0

Shara Lalo, May 25 2018

Row n gives coefficients in expansion of (-1+2x)^n. Row sums=1.

In the center-justified triangle, the numbers in skew diagonals pointing top-Left give the triangle in A133156 (coefficients of Chebyshev polynomials of the second kind), and the numbers in skew diagonals pointing top-right give the triangle in A305098. The coefficients in the expansion of 1/(1-x) are given by the sequence generated by the row sums. The generating function of the central terms is 1/sqrt(1+8x), signed version of A059304.

			Triangle begins:
   1;
  -1,   2;
   1,  -4,   4;
  -1,   6, -12,    8;
   1,  -8,  24,  -32,   16;
  -1,  10, -40,   80,  -80,    32;
   1, -12,  60, -160,  240,  -192,   64;
  -1,  14, -84,  280, -560,   672, -448,   128;
   1, -16, 112, -448, 1120, -1792, 1792, -1024, 256;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 389-391.

Shara Lalo, Skew diagonals in center-justified triangle
Paweł Lorek and Piotr Markowski, Absorption time and absorption probabilities for a family of multidimensional gambler models, arXiv:1812.00690 [math.PR], 2018.

Row sums give A000012.
Signed version of A013609 ((1+2*x)^n).
Cf. A033999 (column 0).

T[0, 0] = 1; T[n_, k_] := If[n < 0 || k < 0, 0, - T[n - 1, k] + 2 T[n - 1, k - 1]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
For[i = 0, i < 4, i++, Print[CoefficientList[Expand[(-1 +2 x)^i], x]]]

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, -T(n-1, k) + 2*T(n-1, k-1)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 26 2018

G.f.: 1 / (1 + t - 2t*x).

T(n,k) = (-1)^(n+k)*2^k*binomial(n,k). - Stefano Spezia, Aug 08 2025

cp's OEIS Frontend

A167591 A triangle related to the a(n) formulas of the rows of the ED4 array A167584.

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A126443 a(n) = Sum_{k=0..n-1} C(n-1,k)*a(k)*2^k for n>0, with a(0)=1.

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A167580 A triangle related to the a(n) formulas of the rows of the ED3 array A167572.

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

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A001848 Crystal ball sequence for 6-dimensional cubic lattice.

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A130123 Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).

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A105728 Triangle read by rows: T(n,1) = 1, T(n,n) = n and for 1 < k < n: T(n,k) = T(n-1,k-1) + 2*T(n-1,k).

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A126443 a(n) = Sum_{k=0..n-1} C(n-1,k)a(k)2^k for n>0, with a(0)=1.

A006588 a(n) = 4^n(3n)!/((2n)!n!).