A030195 -id:A030195 - cp's OEIS frontend

A099093 Riordan array (1, 3+3x).

1, 0, 3, 0, 3, 9, 0, 0, 18, 27, 0, 0, 9, 81, 81, 0, 0, 0, 81, 324, 243, 0, 0, 0, 27, 486, 1215, 729, 0, 0, 0, 0, 324, 2430, 4374, 2187, 0, 0, 0, 0, 81, 2430, 10935, 15309, 6561, 0, 0, 0, 0, 0, 1215, 14580, 45927, 52488, 19683, 0, 0, 0, 0, 0, 243, 10935, 76545, 183708, 177147, 59049

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A030195. Diagonal sums are A099094.
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).
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008

			Rows begin:
  1;
  0, 3;
  0, 3, 9;
  0, 0, 18, 27;
  0, 0, 9, 81, 81;
  0, 0, 0, 81, 324, 243;
  0, 0, 0, 27, 486, 1215, 729;
  ...

Cf. A038221.

[[Binomial(k,n-k)*3^k: k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Feb 21 2015 /* as the triangle */

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(k, n-k)*3^k, ", ");); print(););} \\ Michel Marcus, Feb 21 2015

T(n,k) = binomial(k, n-k)*3^k. - corrected by Michel Marcus, Feb 21 2015
Columns have g.f. (3x+3x^3)^k.
T(n,k) = A026729(n,k)*3^k. - Philippe Deléham, Jul 29 2006

A099581 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k-1)*3^(n-k-1).

Original entry on oeis.org

0, 0, 1, 3, 15, 54, 216, 810, 3105, 11745, 44631, 169128, 641520, 2431944, 9221121, 34959195, 132543135, 502506990, 1905156936, 7222991778, 27384465825, 103822372809, 393620574951, 1492328843280, 5657848431840, 21450531825360

Offset: 0

Paul Barry, Oct 23 2004

In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k-1)*r^(n-k-1) has g.f. x^2/((1-r*x^2)*(1-r*x-r*x^2)) and satisfies a(n) = r*a(n-1) + 2*r*a(n-2) - r^2*a(n-3) - r^2*a(n-4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,6,-9,-9).

Cf. A030195, A099177, A099582.

[n le 4 select Floor((n-1)^2/3) else 3*Self(n-1) +6*Self(n-2) -9*Self(n-3) -9*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 23 2022

LinearRecurrence[{3,6,-9,-9},{0,0,1,3},40] (* Harvey P. Dale, Jun 07 2021 *)

@CachedFunction
def a(n):
    if (n<4): return floor(n^2/3)
    else: return 3*a(n-1) + 6*a(n-2) - 9*a(n-3) - 9*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 23 2022

G.f.: x^2/((1-3*x^2)*(1-3*x-3*x^2)).
a(n) = 3*a(n-1) + 6*a(n-2) - 9*a(n-3) - 9*a(n-4).
From G. C. Greubel, Jul 23 2022: (Start)
a(n) = (2*(-i*sqrt(3))^(n-1)*ChebyshevU(n-1, i*sqrt(3)/2)  - (1-(-1)^n)*3^((n - 1)/2))/6.
E.g.f.: (4*exp(3*x/2)*sinh(sqrt(21)*x/2) - 2*sqrt(7)*sinh(sqrt(3)*x))/(6*sqrt(21)). (End)

A123603 Triangle T(n,k), 0<=k<=n, read by rows, with T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k-1) + T(n-2,k).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 3, 3, 5, 5, 9, 5, 5, 8, 10, 17, 17, 10, 8, 13, 18, 36, 35, 36, 18, 13, 21, 33, 69, 81, 81, 69, 33, 21, 34, 59, 133, 167, 199, 167, 133, 59, 34, 55, 105, 249, 345, 435, 435, 345, 249, 105, 55, 89, 185, 462, 687, 945, 1005, 945, 687, 462, 185, 89

Offset: 0

Philippe Deléham, Nov 14 2006, Mar 14 2014

			Triangle begins:
1;
1, 1;
2, 1, 2;
3, 3, 3, 3;
5, 5, 9, 5, 5;
8, 10, 17, 17, 10, 8;
13, 18, 36, 35, 36, 18, 13;
21, 33, 69, 81, 81, 69, 33, 21;
34, 59, 133, 167, 199, 167, 133, 59, 34;
55, 105, 249, 345, 435, 435, 345, 249, 105, 55;
89, 185, 462, 687, 945, 1005, 945, 687, 462, 185, 89; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000045, A000129, A322239 (central terms).

CoefficientList[CoefficientList[Series[1/(1 - x - x*y - x^2 + x^2*y - x^2*y^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 16 2017 *)
T[0, 0] := 1; T[n_, k_] := If[k < 0 || k > n, 0, T[n - 1, k - 1] + T[n - 1, k] + T[n - 2, k - 2] - T[n - 2, k - 1] + T[n - 2, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Oct 16 2017 *)

T(n,k) = T(n,n-k).
T(n,0) = Fibonacci(n+1) = A000045(n+1).
T(n+1,1) = A010049(n+1).
Sum_{k,0<=k<=n} T(n,k)*x^k = A000045(n+1), A000129(n+1), A030195(n+1), A015532(n+1) for x = 0, 1, 2, 3 respectively.
G.f.: 1/(1 - x - x*y - x^2 + x^2*y - x^2*y^2).

A172375 Triangle T(n, k, q) = c(n-1, q)c(n, q)/(c(k-1, q)^2c(n-k, q)c(n-k+1, q)f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 48, 48, 1, 1, 352, 2816, 352, 1, 1, 2640, 154880, 154880, 2640, 1, 1, 19680, 8659200, 63500800, 8659200, 19680, 1, 1, 146944, 481976320, 26508697600, 26508697600, 481976320, 146944, 1, 1, 1096704, 26859012096, 11012194959360, 82591462195200, 11012194959360, 26859012096, 1096704, 1

Offset: 1

Roger L. Bagula, Feb 01 2010

			Triangle begins as:
  1;
  1,      1;
  1,      6,         1;
  1,     48,        48,           1;
  1,    352,      2816,         352,           1;
  1,   2640,    154880,      154880,        2640,         1;
  1,  19680,   8659200,    63500800,     8659200,     19680,      1;
  1, 146944, 481976320, 26508697600, 26508697600, 481976320, 146944, 1;

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A002605, A030195, this sequence (q=2), A172376 (q=3).

f[n_, q_]:= (-I*Sqrt[q])^(n-1)*ChebyshevU[n-1, I*Sqrt[q]/2];
c[n_, q_]:= Product[f[j, q], {j,n}];
T[n_, k_, q_]:= c[n-1, q]*c[n, q]/(c[k-1, q]^2*c[n-k, q]*c[n-k+1, q]*f[k, q]);
Table[T[n, k, 2], {n,12}, {k, n}]//Flatten (* modified by G. C. Greubel, May 07 2021 *)

@CachedFunction
def f(n,q): return (-i*sqrt(q))^(n-1)*chebyshev_U(n-1, i*sqrt(q)/2)
def c(n,q): return product( f(j,q) for j in (1..n) )
def T(n,k,q): return c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q))
flatten([[T(n,k,2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 07 2021

T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 2.

T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k,q)), where c(n, q) = Product_{j=1..n} f(j, q), and f(n, q) = (-I*sqrt(q))^(n-1)*ChebyshevU(n-1, i*sqrt(q)/2). - G. C. Greubel, May 07 2021

Edited by G. C. Greubel, May 07 2021

A172376 Triangle T(n, k, q) = c(n-1, q)c(n, q)/(c(k-1, q)^2c(n-k, q)c(n-k+1, q)f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 180, 180, 1, 1, 2565, 38475, 2565, 1, 1, 36936, 7895070, 7895070, 36936, 1, 1, 530712, 1633531536, 23277824388, 1633531536, 530712, 1, 1, 7628985, 337399490610, 69234375473172, 69234375473172, 337399490610, 7628985, 1, 1, 109656180, 69713779364775, 205544107079102610, 2959835141939077584, 205544107079102610, 69713779364775, 109656180, 1

Offset: 1

Roger L. Bagula, Feb 01 2010

			Triangle begins as:
  1;
  1,      1;
  1,     12,          1;
  1,    180,        180,           1;
  1,   2565,      38475,        2565,          1;
  1,  36936,    7895070,     7895070,      36936,      1;
  1, 530712, 1633531536, 23277824388, 1633531536, 530712, 1;

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A002605, A030195, A172375 (q=2), this sequence (q=3).

f[n_, q_]:= (-I*Sqrt[q])^(n-1)*ChebyshevU[n-1, I*Sqrt[q]/2];
c[n_, q_]:= Product[f[j, q], {j,n}];
T[n_, k_, q_]:= c[n-1, q]*c[n, q]/(c[k-1, q]^2*c[n-k, q]*c[n-k+1, q]*f[k, q]);
Table[T[n, k, 3], {n,12}, {k, n}]//Flatten (* modified by G. C. Greubel, May 07 2021 *)

@CachedFunction
def f(n,q): return (-i*sqrt(q))^(n-1)*chebyshev_U(n-1, i*sqrt(q)/2)
def c(n,q): return product( f(j,q) for j in (1..n) )
def T(n,k,q): return c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q))
flatten([[T(n,k,3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 07 2021

T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 3.

T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k,q)), where c(n, q) = Product_{j=1..n} f(j, q), and f(n, q) = (-I*sqrt(q))^(n-1)*ChebyshevU(n-1, i*sqrt(q)/2). - G. C. Greubel, May 07 2021

Edited by G. C. Greubel, May 07 2021

A103820 Whitney transform of 3^n.

Original entry on oeis.org

1, 4, 16, 61, 232, 880, 3337, 12652, 47968, 181861, 689488, 2614048, 9910609, 37573972, 142453744, 540083149, 2047610680, 7763081488, 29432076505, 111585473980, 423052651456, 1603914376309, 6080901083296, 23054446378816

Offset: 0

Paul Barry, Feb 16 2005

Partial sums of A030195. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,-3).

Cf. A004070, A030195.

Equals (A108306(n+1) - 1)/5.

I:=[1,4,16]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 18 2017

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-1]+3*a[n-2]+1 od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 14 2008

Join[{a=0,b=1},Table[c=3*b+3*a+1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
LinearRecurrence[{4, 0, -3}, {1, 4, 16}, 40] (* Vincenzo Librandi, Aug 18 2017 *)

G.f.: 1/((1-x)(1-3x-3x^2));
a(n) = 4a(n-1) - 3a(n-3);
a(n) = Sum_{k=0..n} (Sum_{i=0..n} C(k, i-k))*3^k.
a(n) = 3(a(n-1) + a(n-2)) + 1, n > 1. [Gary Detlefs, Jun 21 2010]

A134927 a(0)=a(1)=1; a(n) = 3*(a(n-1) + a(n-2)).

Original entry on oeis.org

1, 1, 6, 21, 81, 306, 1161, 4401, 16686, 63261, 239841, 909306, 3447441, 13070241, 49553046, 187869861, 712268721, 2700415746, 10238053401, 38815407441, 147160382526, 557927369901, 2115263257281, 8019571881546, 30404505416481

Offset: 0

Rolf Pleisch, Jan 29 2008

Joshua Zucker, Feb 23 2008, Table of n, a(n) for n = 0..50
Index entries for linear recurrences with constant coefficients, signature (3,3).

Essentially the same as A108306.

a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-1]+3*a[n-2] od: seq(a[n], n=0..33); # Zerinvary Lajos, Dec 14 2008

Mathematica
```
LinearRecurrence[{3, 3}, {1, 1}, 30]
```

a=[1,1];for(i=2,10,a=concat(a,3*a[#a]+3*a[#a-1]));a \\ Charles R Greathouse IV, Oct 04 2011

from sage.combinat.sloane_functions import recur_gen2
it = recur_gen2(1,1,3,3)
[next(it) for i in range(25)] # Zerinvary Lajos, Jun 25 2008

From R. J. Mathar, Jan 31 2008: (Start)
O.g.f.: (-1+2*x)/(-1 + 3*x + 3*x^2).
a(n) = A030195(n+1)-2*A030195(n). (End)
a(n) = A108306(n-1), n>0. - R. J. Mathar, Oct 04 2011
a(n) ~ 3.7912878474...^n, where the constant is A090458. - Charles R Greathouse IV, Oct 04 2011

More terms from Joshua Zucker, Feb 23 2008

A172012 Expansion of (2-3x)/(1-3x-3*x^2) .

Original entry on oeis.org

2, 3, 15, 54, 207, 783, 2970, 11259, 42687, 161838, 613575, 2326239, 8819442, 33437043, 126769455, 480619494, 1822166847, 6908359023, 26191577610, 99299809899, 376474162527, 1427321917278, 5411388239415, 20516130470079, 77782556128482, 294896059795683

Offset: 0

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010

The case k=3 in a family of sequences a(n) = L(k,n), L(k,n)=k*(L(k,n-1)+L(k,n-2)), L(k,0)=2 and L(k,1)=k.

The case k=1 is A000032 (classic Lucas sequence), k=2 is A080040, this here is essentially A085480.

Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (3,3).

CoefficientList[Series[(2-3x)/(1-3x-3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{3,3},{2,3},31] (* Harvey P. Dale, Aug 24 2011 *)

a(n) = 3*( a(n-1)+a(n-2) ) = 2*A030195(n+1)-3*A030195(n).
L(k,n) = c^n+b^n where c=(k+d)/2 ; b=(k-d)/2; d=sqrt(k*(k+4)) (Binet formula).
a(0)=2, a(1)=3, a(n) = 3*a(n-1)+3*a(n-2). [Harvey P. Dale, Aug 24 2011]
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 21*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

Edited and extended by R. J. Mathar, Jan 23 2010

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 3, 5, 1, -1, 5, 10, 4, -2, -1, 8, 20, 12, -4, -4, 0, 13, 38, 31, -4, -13, -2, 1, 21, 71, 73, 3, -33, -11, 3, 1, 34, 130, 162, 34, -74, -42, 6, 6, 0, 55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Offset: 0

Philippe Deléham, Dec 06 2011

Row-reversed variant of A123585. Row sums: 2^n.

			Triangle begins:
1
1, 1
2, 2, 0
3, 5, 1, -1
5, 10, 4, -2, -1
8, 20, 12, -4, -4, 0
13, 38, 31, -4, -13, -2, 1
21, 71, 73, 3, -33, -11, 3, 1
34, 130, 162, 34, -74, -42, 6, 6, 0
55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Cf. Columns: A000045, A001629, A129707.

Diagonals: A010892, A099254, Antidiagonal sums: A158943.

G.f.: 1/(1-(1+y)*x+(y+1)*(y-1)*x^2).
T(n,0) = A000045(n+1).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0)= 1 and T(n,k)= 0 if nSum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A090591(n), (-1)^n*A106852(n), A000007(n), A000045(n+1), A000079(n), A057083(n), A190966(n+1) for n = -3, -2, -1, 0, 1, 2, 3 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A010892(n), A000079(n), A030195(n+1), A180222(n+2) for x = 0, 1, 2, 3 respectively.

A238941 Triangle T(n,k), read by rows given by (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 8, 4, 1, 13, 21, 13, 6, 1, 34, 55, 40, 25, 7, 1, 89, 144, 120, 90, 33, 9, 1, 233, 377, 354, 300, 132, 51, 10, 1, 610, 987, 1031, 954, 483, 234, 62, 12, 1, 1597, 2584, 2972, 2939, 1671, 951, 308, 86, 13, 1, 4181, 6765, 8495, 8850, 5561, 3573, 1345, 480, 100, 15, 1

Offset: 0

Philippe Deléham, Mar 07 2014

Row sums are A025192(n).

			Triangle begins:
1;
1, 1;
2, 3, 1;
5, 8, 4, 1;
13, 21, 13, 6, 1;
34, 55, 40, 25, 7, 1;
89, 144, 120, 90, 33, 9, 1;
233, 377, 354, 300, 132, 51, 10, 1;

Indranil Ghosh, Rows 0..100, flattened

Cf. Columns: A001519, A001906, A238846, A001871.

Cf. Diagonals: A000012, A032766.

nmax=10; Column[CoefficientList[Series[CoefficientList[Series[(1 - 2*x + x*y)/(1 - 3*x + x^2 - x^2*y^2), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 14 2017 *)

G.f. for the column k: x^k*(1-2*x)^A059841(k)/(1-3*x+x^2)^A008619(k).
G.f.: (1-2*x+x*y)/(1-3*x+x^2-x^2*y^2).
T(n,k) = 3*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000007(n), A001519(n), A025192(n), A030195(n+1) for x = -1, 0, 1, 2 respectively.
Sum_{k = 0..n} T(n,k)*3^k = A015525(n) + A015525(n+1).

Data section corrected and extended by Indranil Ghosh, Mar 14 2017

Previous Showing 41-50 of 52 results. Next

cp's OEIS Frontend

A099093 Riordan array (1, 3+3x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

PARI

Formula

A099581 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k-1)*3^(n-k-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A123603 Triangle T(n,k), 0<=k<=n, read by rows, with T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k-1) + T(n-2,k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A172375 Triangle T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 2, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A172376 Triangle T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 3, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A103820 Whitney transform of 3^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A134927 a(0)=a(1)=1; a(n) = 3*(a(n-1) + a(n-2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A172375 Triangle T(n, k, q) = c(n-1, q)c(n, q)/(c(k-1, q)^2c(n-k, q)c(n-k+1, q)f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 2, read by rows.

A172376 Triangle T(n, k, q) = c(n-1, q)c(n, q)/(c(k-1, q)^2c(n-k, q)c(n-k+1, q)f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 3, read by rows.

A172012 Expansion of (2-3x)/(1-3x-3*x^2) .