A008287 -id:A008287 - cp's OEIS frontend

A177882 Trisection of A001317.

1, 15, 85, 771, 4369, 65535, 327685, 3342387, 16843009, 252645135, 1431655765, 12884901891, 73014444049, 1095216660735, 5519032976645, 56294136361779, 281479271743489, 4222189076152335, 23925738098196565

Offset: 0

Vladimir Shevelev, Dec 14 2010

nonn

For n>=1, all terms are in A001969.

Or rows of triangle A008287 mod 2 converted to decimal.

Gheorghe Coserea, Table of n, a(n) for n = 0..200

Cf. A001317, A001969, A005190, A007318, A008287.

f[n_] := BitXor[n, BitShiftLeft[n, 1]]; Table[Nest[f, 1, x], {x, 0, 54, 3}]

a(n) = subst(lift(Pol(Mod([1, 1, 1, 1], 2), 'x)^n), 'x, 2);
vector(19, n, a(n-1))  \\ Gheorghe Coserea, Jun 12 2016

def A177882(n): return sum((bool(~(3*n)&3*n-k)^1)<Chai Wah Wu, May 02 2023

a(n) = A001317(3*n).

Definition rewritten by N. J. A. Sloane, Jan 01 2011

A120987 Triangle read by rows: T(n,k) is the number of ternary words of length n with k strictly increasing runs (0 <= k <= n; for example, the ternary word 2|01|12|02|1|1|012|2 has 8 strictly increasing runs).

Original entry on oeis.org

1, 0, 3, 0, 3, 6, 0, 1, 16, 10, 0, 0, 15, 51, 15, 0, 0, 6, 90, 126, 21, 0, 0, 1, 77, 357, 266, 28, 0, 0, 0, 36, 504, 1107, 504, 36, 0, 0, 0, 9, 414, 2304, 2907, 882, 45, 0, 0, 0, 1, 210, 2850, 8350, 6765, 1452, 55, 0, 0, 0, 0, 66, 2277, 14355, 25653, 14355, 2277, 66, 0, 0, 0, 0, 12

Offset: 0

Emeric Deutsch, Jul 23 2006

Sum of entries in row n is 3^n (A000244).
Sum of entries in column k is A099464(k+1) (a trisection of the tribonacci numbers).
Row n contains 1 + floor(2n/3) nonzero terms.
T(n,n) = (n+1)*(n+2)/2 (the triangular numbers (A000217)).
Sum_{k=0..n} k*T(n,k) = (2n+1)*3^(n-1) = 3*A081038(n-1) for n >= 1.
T(n,k) = A120987(n,n-k).

			T(5,2) = 6 because we have 012|01, 012|02, 012|12, 01|012, 02|012 and 12|012 (the runs are separated by |).
Triangle starts:
  1;
  0,   3;
  0,   3,   6;
  0,   1,  16,  10;
  0,   0,  15,  51,  15;
  0,   0,   6,  90, 126,  21;

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 24, p. 154.

G. C. Greubel, Table of n, a(n) for n = 0..11475
Giuliano Cabrele, Words Partitioned according to Number of Strictly Increasing Runs
MathPages, Balls In Bins With Limited Capacity.

Cf. A000244, A008287, A081038, A099464, A119900, A120987, A265644.

Nb(s,2,q) = A027907(q,s). - Giuliano Cabrele, Dec 11 2015

G:=1/(1-3*t*z-3*t*(1-t)*z^2-t*(1-t)^2*z^3): Gser:=simplify(series(G,z=0,33)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

Flatten[Table[Sum[(-1)^j*Binomial[n + 1, j]*Binomial[3 k - 3 j, n], {j, 0, k}], {n, 0, 10}, {k, 0, n}]] (* G. C. Greubel, Dec 20 2015 *)

// binomial c. defined as in linked document
Cb:=(x,m)->if(0<=m and is(m in Z), binomial(x,m), 0):
// closed formula derived and proved in the linked document
// Qsc(r,q,m) with r=2
T(n,k):=(n,k)->_plus((-1)^(k-j)*Cb(n+1,k-j)*Cb(3*j, n)$j=0..k):
// Giuliano Cabrele, Dec 11 2015

T(n,k) = trinomial(n+1,3n-3k+2) = trinomial(n+1,3k-n) (conjecture).
G.f.: 1/(1-3tz-3t(1-t)z^2-t(1-t)^2*z^3).
Can anyone prove the conjecture (either from the g.f. or combinatorially from the definition)?
From Giuliano Cabrele, Mar 02 2008: (Start)
The conjecture is compatible with the g.f., which can be rewritten as (1-t)/(1-t(1+(1-t)z)^3) and expanded to give T(n,k) = Sum_{j=0..k} (-1)^(k-j)*C(3j, n)*C(n+1, k-j) = Sum_{j=0..k} (-1)^j*C(n+1,j)*C(3k-3j,n) = trinomial(n+1,3k-n) = A027907(n+1,3k-n).
Also (1-t)/(1-t(1+(1-t)z)^2) equals the g.f. for the case of binary words, A119900, where Sum_{j=0..k} (-1)^(k-j)*C(2j,n)*C(n+1,k-j) = C(n+1,2k-n). Changing the exponent to 1 gives 1/(1-zt), the g.f. for the case of unary words, the expansion coefficients of which can be written as Kronecker delta(k-n)^(n+1) = Sum_{j=0..k} (-1)^(k-j)*C(j, n)*C(n+1,k-j).
So the conjecture shifts to that the g.f. is (1-t)/(1-t(1+(1-t)z)^m) and coefficients T(m,n,k) = Sum_{j=0..k} (-1)^(k-j)*C(mj,n)*C(n+1, k-j) may apply to the general case of m-ary words. (End)
Sum_{k=0..n} T(n,k) *(-1)^(n-k) = A157241(n+1). - Philippe Deléham, Oct 25 2011
The generalized conjecture above can in fact be proved, as described in the file "Words Partitioned according to Number of Strictly Increasing Runs" linked above. - Giuliano Cabrele, Dec 11 2015

A349813 Triangle read by rows: row 1 is [3]; for n >= 1, row n gives coefficients of expansion of (-3 - x + x^2 + 3x^3)(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

Original entry on oeis.org

3, -3, -1, 1, 3, -3, -4, -3, 0, 3, 4, 3, -3, -7, -10, -10, -4, 4, 10, 10, 7, 3, -3, -10, -20, -30, -31, -20, 0, 20, 31, 30, 20, 10, 3, -3, -13, -33, -63, -91, -101, -81, -31, 31, 81, 101, 91, 63, 33, 13, 3, -3, -16, -49, -112, -200, -288, -336, -304, -182, 0, 182, 304, 336, 288, 200, 112, 49, 16, 3

Offset: 0

N. J. A. Sloane, Dec 23 2021

The row polynomials can be further factorized, since -3 - x + x^2 + 3*x^3 = -(1-x)*(3 + 4*x + 3*x^2) and 1 + x + x^2 + x^3 = (1+x)*(1+x^2).

The rule for constructing this triangle (ignoring row 0) is the same as that for A008287: each number is the sum of the four numbers immediately above it in the previous row. Here row 1 is [-3, -1, 1, 3] instead of [1, 1, 1, 1].

			Triangle begins:
   3;
  -3,  -1,   1,   3;
  -3,  -4,  -3,   0,   3,    4,   3;
  -3,  -7, -10, -10,  -4,    4,  10,  10,  7,  3;
  -3, -10, -20, -30, -31,  -20,   0,  20, 31, 30,  20, 10,  3;
  -3, -13, -33, -63, -91, -101, -81, -31, 31, 81, 101, 91, 63, 33, 13, 3;
  ...

Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]

Cf. A007318, A008287, A349812, A349815, A349819.

The right half of the triangle gives A349814.

t1:=-3-x+x^2+3*x^3;
m:=1+x+x^2+x^3;
lprint([3]);
for n from 1 to 12 do
w1:=expand(t1*m^(n-1));
w4:=series(w1,x,3*n+1);
w5:=seriestolist(w4);
lprint(w5);
od:

A005719 Quadrinomial coefficients.

Original entry on oeis.org

2, 12, 40, 101, 216, 413, 728, 1206, 1902, 2882, 4224, 6019, 8372, 11403, 15248, 20060, 26010, 33288, 42104, 52689, 65296, 80201, 97704, 118130, 141830, 169182, 200592, 236495, 277356, 323671, 375968, 434808, 500786, 574532, 656712, 748029, 849224, 961077

Offset: 2

N. J. A. Sloane

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, 1987
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

a(n)= A008287(n, 5), n >= 2 (sixth column of quadrinomial coefficients).

A005719:=(2-2*z**2+z**3)/(z-1)**6; [Conjectured by Simon Plouffe in his 1992 dissertation.]

a(n)= binomial(n, 2)*(n^3+11*n^2+46*n-24)/60, n >= 2.
G.f.: (x^2)*(2-2*x^2+x^3)/(1-x)^6. (numerator polynomial is N4(5, x) from A063421.)
a(n) = 2*binomial(n,2) + 6*binomial(n,3) + 4*binomial(n,4) + binomial(n,5) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A005720 Quadrinomial coefficients.

Original entry on oeis.org

1, 10, 44, 135, 336, 728, 1428, 2598, 4455, 7282, 11440, 17381, 25662, 36960, 52088, 72012, 97869, 130986, 172900, 225379, 290444, 370392, 467820, 585650, 727155, 895986, 1096200, 1332289, 1609210, 1932416, 2307888, 2742168, 3242393, 3816330, 4472412, 5219775

Offset: 2

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, 1987
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
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

a(n)= A008287(n, 6), n >= 2 (seventh column of quadrinomial coefficients).

A005720:=-(1+3*z-5*z**2+2*z**3)/(z-1)**7; [Conjectured by Simon Plouffe in his 1992 dissertation.]

LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,44,135,336,728,1428},40] (* or *) Table[Binomial[n+1,3] (n^3+15n^2+86n-120)/120,{n,2,41}] (* Harvey P. Dale, Jun 23 2011 *)

a(n)=(n^6 + 15*n^5 + 85*n^4 - 135*n^3 - 86*n^2 + 120*n)/720 \\ Charles R Greathouse IV, Jun 23 2011

a(n)= binomial(n+1, 3)*(n^3+15*n^2+86*n-120)/120, n >= 2.
G.f.: (x^2)*(1+3*x-5*x^2+2*x^3)/(1-x)^7. (numerator polynomial is N4(6, x) from A063421).
a(0)=1, a(1)=10, a(2)=44, a(3)=135, a(4)=336, a(5)=728, a(6)=1428, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Jun 23 2011
a(n) = binomial(n,2) + 7*binomial(n,3) + 10*binomial(n,4) + 5*binomial(n,5) + binomial(n,6) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A163181 T(n,k) is the number of weak compositions of k into n parts no greater than (n-1) for n>=1, 0<=k<=n(n-1).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 252, 456, 756, 1161, 1666, 2247, 2856, 3431, 3906, 4221, 4332, 4221, 3906, 3431

Offset: 1

Geoffrey Critzer, Jul 22 2009

T(n,k) is the number of length n sequences on an alphabet of {0,1,2,...,n-1} that have a sum of k. Equivalently T(n,k) is the number of functions f:{1,2,...,n}->{0,1,2,...,n-1} such that Sum(f(i)=k, i=1...n).

Row n is also row n of the array of q-nomial coefficients. - Matthew Vandermast, Oct 31 2010

			T(3,4) = 6 because there are 6 ternary sequences of length three that sum to 4: [0, 2, 2], [1, 1, 2], [1, 2, 1], [2, 0, 2], [2, 1, 1], [2, 2, 0].

Alois P. Heinz, Rows n = 1..32, flattened

The maximum of row n is in column k=n(n-1)/2 = A000217(n-1).

For q-nomial arrays, see A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890. See also A181567. - Matthew Vandermast, Oct 31 2010

b:= proc(n, k, l) option remember; `if`(k=0, 1,
      `if`(l=0, 0, add(b(n, k-j, l-1), j=0..min(n-1, k))))
    end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(n, k), k=0..n*(n-1)), n=1..8);  # Alois P. Heinz, Feb 21 2013

(*warning very inefficient*) Table[Distribution[Map[Total, Strings[Range[n], n]]], {n, 1, 6}]//Grid
nn=100;Table[CoefficientList[Series[Sum[x^i,{i,0,n-1}]^n,{x,0,nn}],x],{n,1,10}]//Grid (* Geoffrey Critzer, Feb 21 2013*)

O.g.f. for row n is ((1-x^n)/(1-x))^n. For k<=(n-1), T(n,k) = C(n+k-1,k).

A349815 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1 - x + x^2 + x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

Original entry on oeis.org

1, -1, -1, 1, 1, -1, -2, -1, 0, 1, 2, 1, -1, -3, -4, -4, -2, 2, 4, 4, 3, 1, -1, -4, -8, -12, -13, -8, 0, 8, 13, 12, 8, 4, 1, -1, -5, -13, -25, -37, -41, -33, -13, 13, 33, 41, 37, 25, 13, 5, 1, -1, -6, -19, -44, -80, -116, -136, -124, -74, 0, 74, 124, 136, 116, 80, 44, 19, 6, 1, -1, -7, -26, -70, -149, -259, -376, -456, -450, -334, -124, 124, 334, 450, 456, 376, 259, 149, 70, 26, 7, 1

Offset: 0

N. J. A. Sloane, Dec 23 2021

The row polynomials can be further factorized, since -3 - x + x^2 + 3*x^3 = -(1-x)*(1+x)^2 and 1 + x + x^2 + x^3 = (1+x)*(1+x^2).
The rule for constructing this triangle (ignoring row 0) is the same as that for A008287: each number is the sum of the four numbers immediately above it in the previous row. Here row 1 is [-1, -1, 1, 3] instead of [1, 1, 1, 1].
This is a companion to A008287 and A349813.

			Triangle begins:
   1;
  -1, -1,   1,   1;
  -1, -2,  -1,   0,   1,   2,   1;
  -1, -3,  -4,  -4,  -2,   2,   4,   4,  3,  1;
  -1, -4,  -8, -12, -13,  -8,   0,   8, 13, 12,  8,  4,  1;
  -1, -5, -13, -25, -37, -41, -33, -13, 13, 33, 41, 37, 25, 13, 5, 1;
  ...

Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]

Cf. A007318, A008287, A349812, A349813.

The right half of the triangle gives A349816. For the central nonzero entries see A349818.

t1:=-1-x+x^2+x^3;
m:=1+x+x^2+x^3;
lprint([3]);
for n from 1 to 12 do
w1:=expand(t1*m^(n-1));
w4:=series(w1,x,3*n+1);
w5:=seriestolist(w4);
lprint(w5);
od:

A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 19, 4, 1, 1, 70, 141, 44, 5, 1, 1, 252, 1107, 580, 85, 6, 1, 1, 924, 8953, 8092, 1751, 146, 7, 1, 1, 3432, 73789, 116304, 38165, 4332, 231, 8, 1, 1, 12870, 616227, 1703636, 856945, 135954, 9331, 344, 9, 1, 1, 48620, 5196627, 25288120, 19611175, 4395456, 398567, 18152, 489, 10, 1

Offset: 0

Stefano Spezia, Dec 06 2021

			The array begins:
n\s |   0     1     2     3     4
----+----------------------------
  0 |   1     1     1     1     1 ...
  1 |   1     2     3     4     5 ...
  2 |   1     6    19    44    85 ...
  3 |   1    20   141   580  1751 ...
  4 |   1    70  1107  8092 38165 ...
  ...

William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 2.

Cf. A000984 (s=1), A082758 (s=2), A005721 (s=3), A349936 (s=4), A063419 (s=5), A270918 (n=s), A163269 (s>0).
Cf. A007318, A008287, A025012, A027907, A035343, A063260.
Cf. A349934, A349935.

T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; A[n_,s_]:=T[2n,s n,s]; Flatten[Table[A[n-s,s],{n,0,9},{s,0,n}]]

A(n, s) = T(2*n, s*n, s), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.

A005726 Quadrinomial coefficients.

Original entry on oeis.org

1, 2, 6, 20, 65, 216, 728, 2472, 8451, 29050, 100298, 347568, 1208220, 4211312, 14712960, 51507280, 180642391, 634551606, 2232223626, 7862669700, 27727507521, 97884558992, 345891702456, 1223358393120, 4330360551700

Offset: 1

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. K. Guy, Letter to N. J. A. Sloane, 1987

for n from 1 to 40 do printf(`%d,`,coeff(expand(sum(x^j, j=0..3)^n), x, n-1)) od:
F := (t-1)^2*(t^2+1)^2/(2*t^3-t^2+1);  G := t/((t-1)*(t^2+1)); Ginv := RootOf(numer(G-x),t);  ogf := series(eval(F,t=Ginv),x=0,20); # Mark van Hoeij, Oct 30 2011

Table[Sum[Binomial[n,k]Binomial[n,2k+1],{k,0,Floor[n/2]}],{n,30}] (* Harvey P. Dale, Oct 19 2013 *)

a(n) = Sum_{k=0..floor(n/2)}, C(n,k) C(n,2k+1). - Paul Barry, May 15 2003
a(n) = Sum[(-1)^k binomial[n,k] binomial[2n-2-4k,n-1],{k,0,Floor[(n-1)/4]}]. - David Callan, Jul 03 2006
G.f.: F(G^(-1)(x)) where F(t) = (t-1)^2*(t^2+1)^2/(2*t^3-t^2+1) and G(t) = t/((t-1)*(t^2+1)). - Mark van Hoeij, Oct 30 2011
Conjecture: 2*(n-1)*(2*n+1)*(13*n-14)*a(n) +(-143*n^3+297*n^2-148*n+12) *a(n-1) -4*(n-1)*(26*n^2-41*n+9)*a(n-2) -16*(n-1)*(n-2)*(13*n-1) *a(n-3)=0. - R. J. Mathar, Nov 13 2012
a(n) = A008287(n,n-1). - Sean A. Irvine, Aug 15 2016

More terms from James Sellers, Aug 21 2000

A064055 Ninth column of quadrinomial coefficients.

Original entry on oeis.org

3, 31, 155, 546, 1554, 3823, 8451, 17205, 32802, 59268, 102388, 170261, 273975, 428418, 653242, 973998, 1423461, 2043165, 2885169, 4014076, 5509328, 7467801, 10006725, 13266955, 17416620, 22655178, 29217906

Offset: 0

Wolfdieter Lang, Aug 29 2001

A001919 (eighth column).

Table[3Binomial[n+3,3]+19Binomial[n+3,4]+30Binomial[n+3,5]+21 Binomial[n+3,6]+ 7 Binomial[n+3,7]+ Binomial[n+3,8],{n,0,30}] (* Harvey P. Dale, Apr 30 2022 *)

a(n)= A008287(n+3, 8)= binomial(n+3, 3)*(n^5+46*n^4+875*n^3+7118*n^2+23880*n+20160)/(8!/3!), n >= 0.
G.f.: (3+4*x-16*x^2+15*x^3-6*x^4+x^5 )/(1-x)^9, numerator polynomial is N4(8, x) from the array A063421.
a(n) = 3*C(n+3,3) + 19*C(n+3,4) + 30*C(n+3,5) + 21*C(n+3,6) + 7*C(n+3,7) + C(n+3,8) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

cp's OEIS Frontend

A177882 Trisection of A001317.

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A120987 Triangle read by rows: T(n,k) is the number of ternary words of length n with k strictly increasing runs (0 <= k <= n; for example, the ternary word 2|01|12|02|1|1|012|2 has 8 strictly increasing runs).

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A349813 Triangle read by rows: row 1 is [3]; for n >= 1, row n gives coefficients of expansion of (-3 - x + x^2 + 3*x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

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A005719 Quadrinomial coefficients.

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A005720 Quadrinomial coefficients.

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A163181 T(n,k) is the number of weak compositions of k into n parts no greater than (n-1) for n>=1, 0<=k<=n(n-1).

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A349815 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1 - x + x^2 + x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

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A349813 Triangle read by rows: row 1 is [3]; for n >= 1, row n gives coefficients of expansion of (-3 - x + x^2 + 3x^3)(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.