A037027 -id:A037027 - cp's OEIS frontend

A055830 Triangle T read by rows: diagonal differences of triangle A037027.

1, 1, 0, 2, 1, 0, 3, 3, 1, 0, 5, 7, 4, 1, 0, 8, 15, 12, 5, 1, 0, 13, 30, 31, 18, 6, 1, 0, 21, 58, 73, 54, 25, 7, 1, 0, 34, 109, 162, 145, 85, 33, 8, 1, 0, 55, 201, 344, 361, 255, 125, 42, 9, 1, 0, 89, 365, 707, 850, 701, 413, 175, 52, 10, 1, 0, 144, 655, 1416, 1918, 1806, 1239, 630, 236, 63, 11, 1, 0

Offset: 0

Clark Kimberling, May 28 2000

Or, coefficients of a generalized Lucas-Pell polynomial read by rows. - Philippe Deléham, Nov 05 2006

Equals A046854(shifted) * Pascal's triangle; where A046854 is shifted down one row and "1" inserted at (0,0). - Gary W. Adamson, Dec 24 2008

			Triangle begins:
   1
   1,   0
   2,   1,   0
   3,   3,   1,   0
   5,   7,   4,   1,   0
   8,  15,  12,   5,   1,   0
  13,  30,  31,  18,   6,   1,  0
  21,  58,  73,  54,  25,   7,  1, 0
  34, 109, 162, 145,  85,  33,  8, 1, 0
  55, 201, 344, 361, 255, 125, 42, 9, 1, 0
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 3D.

Left-hand columns include A000045, A023610.
Right-hand columns include A055831, A055832, A055833, A055834, A055835, A055836, A055837, A055838, A055839, A055840.
Row sums: A001333 (numerators of continued fraction convergents to sqrt(2)).
Cf. A122075 (another version).
Cf. A046854. - Gary W. Adamson, Dec 24 2008

function T(n,k)
  if k lt 0 or k gt n then return 0;
  elif k eq 0 then return Fibonacci(n+1);
  elif n eq 1 and k eq 1 then return 0;
  else return T(n-1,k-1) + T(n-1,k) + T(n-2,k);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 21 2020

with(combinat);
T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then fibonacci(n+1)
    elif n=1 and k=1 then 0
    else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Jan 21 2020

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 19 2017 *)

T(n,k) = if(k<0 || k>n, 0, if(k==0, fibonacci(n+1), if(n==1 && k==1, 0, T(n-1, k-1) + T(n-1, k) + T(n-2, k) )));
for(n=0,12, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 21 2020

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return fibonacci(n+1)
    elif (n==1 and k==1): return 0
    else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 21 2020

G.f.: (1-y*z) / (1-y*(1+y+z)).
T(i, j) = R(i-j, j), where R(0, 0)=1, R(0, j)=0 for j >= 1, R(1, j)=1 for j >= 0, R(i, j) = Sum_{k=0..j} (R(i-2, k) + R(i-1, k)) for i >= 1, j >= 1.
Sum_{k=0..n} x^k*T(n,k) = A039834(n-2), A000012(n), A000045(n+1), A001333(n), A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), A015455(n), A015456(n), A015457(n) for x= -2,-1,0,1,2,3,4,5,6,7,8,9,10. - Philippe Deléham, Oct 22 2006
Sum_{k=0..floor(n/2)} T(n-k,k) = A011782(n). - Philippe Deléham, Oct 22 2006
Triangle T(n,k), 0 <= k <= n, given by [1, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 05 2006
T(n,0) = Fibonacci(n+1) = A000045(n+1). Sum_{k=0..n} T(n,k) = A001333(n). T(n,k)=0 if k > n or if k < 0, T(0,0)=1, T(1,1)=0, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k). - Philippe Deléham, Nov 05 2006

Edited by Ralf Stephan, Jan 12 2005

A111006 Another version of Fibonacci-Pascal triangle A037027.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 5, 0, 0, 0, 3, 10, 8, 0, 0, 0, 1, 9, 20, 13, 0, 0, 0, 0, 4, 22, 38, 21, 0, 0, 0, 0, 1, 14, 51, 71, 34, 0, 0, 0, 0, 0, 5, 40, 111, 130, 55, 0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89, 0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144

Offset: 0

Philippe Deléham, Oct 02 2005

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 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.
Row sums are the Jacobsthal numbers A001045(n+1) and column sums form Pell numbers A000129.
Maximal column entries: A038149 = {1, 1, 2, 5, 10, 22, ...}.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, ...).
Triangle read by rows: T(n,n-k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)). - Philippe Deléham, Feb 17 2014
Diagonal sums are A013979(n). - Philippe Deléham, Feb 17 2014
T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and 1 X 2 tiles. - Emeric Deutsch, Aug 14 2014

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 1, 5,  5;
  0, 0, 0, 3, 10,  8;
  0, 0, 0, 1,  9, 20, 13;
  0, 0, 0, 0,  4, 22, 38,  21;
  0, 0, 0, 0,  1, 14, 51,  71,  34;
  0, 0, 0, 0,  0,  5, 40, 111, 130,  55;
  0, 0, 0, 0,  0,  1, 20, 105, 233, 235,  89;
  0, 0, 0, 0,  0,  0,  6,  65, 256, 474, 420, 144;

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000045, A000129, A001045, A037027, A038112, A038149, A084938, A128100 (reversed version).

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A114197, A162741, A228074.

a111006 n k = a111006_tabl !! n !! k
a111006_row n = a111006_tabl !! n
a111006_tabl =  map fst $ iterate (\(us, vs) ->
   (vs, zipWith (+) (zipWith (+) ([0] ++ us ++ [0]) ([0,0] ++ us))
                    ([0] ++ vs))) ([1], [0,1])
-- Reinhard Zumkeller, Aug 15 2013

T(0, 0) = 1, T(n, k) = 0 for k < 0 or for n < k, T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).
T(n, k) = A037027(k, n-k). T(n, n) = A000045(n+1). T(3n, 2n) = (n+1)*A001002(n+1) = A038112(n).
G.f.: 1/(1-yx(1-x)-x^2*y^2). - Paul Barry, Oct 04 2005
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A053524(n+1), (-1)^n*A083858(n+1), (-1)^n*A002605(n), A033999(n), A000007(n), A001045(n+1), A083099(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 02 2006
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 respectively. - Philippe Deléham, Feb 17 2014

A038137 Reflection of A037027: T(n,m) = U(n,n-m), m=0..n, where U is as in A037027.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 9, 10, 5, 1, 5, 14, 22, 20, 8, 1, 6, 20, 40, 51, 38, 13, 1, 7, 27, 65, 105, 111, 71, 21, 1, 8, 35, 98, 190, 256, 233, 130, 34, 1, 9, 44, 140, 315, 511, 594, 474, 235, 55, 1, 10, 54, 192, 490, 924, 1295, 1324, 942, 420, 89, 1, 11, 65, 255

Offset: 0

Floor van Lamoen

Number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (2,2). - Joerg Arndt, Jul 01 2011
The n-th diagonal D(n) = {T(n,0), T(n+1,1), ..., T(n+m,m), ...} of the triangle has generating function F(x) = 1/(1 - x - x^2)^(n+1) for n = 0,1,2,.... - L. Edson Jeffery, Mar 20 2011
Let p(n,x) denote the Fibonacci polynomial, defined by p(1,x) = 1, p(2,x) = x, and p(n,x) = x*p(n-1,x) + p(n-2,x) for n >= 3. Let q(n,x) be the numerator polynomial of the rational function p(n, 1 + 1/x). The coefficients of the polynomial q(n,x) are given by the (n-1)-th row of T(n,k). E.g., p(5,x) = 1 + 3*x^2 + x^4 gives q(5,x) = 1 + 4*x + 9*x^2 + 10*x^2 + 5*x^4. - Clark Kimberling, Nov 04 2013

			Triangle T(n,k) (with rows n >= 0 and columns 0 <= k <= n) begins
  1;
  1, 1;
  1, 2,  2;
  1, 3,  5,  3;
  1, 4,  9, 10,   5;
  1, 5, 14, 22,  20,   8;
  1, 6, 20, 40,  51,  38, 13;
  1, 7, 27, 65, 105, 111, 71, 21;
  ...

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Pieter Moree, Convoluted convolved Fibonacci numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
Eric Weisstein's World of Mathematics, Fibonacci polynomial.

Row sums are Pell numbers A000129.
Diagonal sums are unsigned version of A077930.
Cf. A037027, A059841, A208336.

a038137 n k = a038137_tabl !! n !! k
a038137_row n = a038137_tabl !! n
a038137_tabl = map reverse a037027_tabl
-- Reinhard Zumkeller, Jul 08 2012

From Paul Barry, Oct 24 2005: (Start)
G.f.: 1/(1 - x - x*y - x^2*y^2).
T(n,k) = Sum_{j=0..n} C((n+j)/2, j) * (1 + (-1)^(n+j)) * C(j, n-k)/2. (End)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(n,k) = 0 if n < 0 or if n < k, and T(0,0) = 1. - Philippe Deléham, Nov 30 2006
Sum_{k=0..n} (-1)^k*T(n,k) = A059841(n). - Philippe Deléham, Nov 30 2006
T(n,k) = A208336(n+1,k).- Philippe Deléham, Apr 05 2012

Title corrected by L. Edson Jeffery, Apr 23 2011

Corrected by Philippe Deléham, Apr 05 2012

A060920 Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 13, 20, 9, 1, 34, 71, 51, 14, 1, 89, 235, 233, 105, 20, 1, 233, 744, 942, 594, 190, 27, 1, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 1597, 6865, 12473, 12402, 7285, 2534, 490, 44, 1, 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1

Offset: 0

Wolfdieter Lang, Apr 20 2001

Companion triangle (odd-indexed members) A060921.

			Triangle begins as:
      1;
      2,     1;
      5,     5,      1;
     13,    20,      9,      1;
     34,    71,     51,     14,      1;
     89,   235,    233,    105,     20,     1;
    233,   744,    942,    594,    190,    27,     1;
    610,  2285,   3522,   2860,   1295,   315,    35,    1;
   1597,  6865,  12473,  12402,   7285,  2534,   490,   44,    1;
   4181, 20284,  42447,  49963,  36122, 16407,  4578,  726,   54,  1;
  10946, 59155, 140109, 190570, 163730, 91959, 33705, 7776, 1035, 65, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, Nov. 2005, pp. 359-370.

Column sequences: A001519 (k=0), A054444 (k=1), A061178 (k=2), A061179 (k=3), A061180 (k=4), A061181 (k=5).

Cf. A000045, A001519, A007583, A037027, A060921, A061176.

A060920:= func< n,k | (&+[Binomial(2*n-k-j, j)*Binomial(2*n-k-2*j, k): j in [0..2*n-k]]) >;
[A060920(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 06 2021

A060920[n_, k_]:= Sum[Binomial[2*n-k-j, j]*Binomial[2*n-k-2*j, k], {j,0,2*n-k}];
Table[A060920[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)

def A060920(n,k): return sum(binomial(2*n-k-j, j)*binomial(2*n-k-2*j, k) for j in (0..2*n-k))
flatten([[A060920(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 06 2021

T(n, k) = A037027(2*n-k, k).
T(n, k) = ((2*(n-k) + 1)*A060921(n-1, k-1) + 4*n*T(n-1, k-1))/(5*k), n >= k >= 1.
T(n, 0) = F(n)^2 + F(n+1)^2 = A001519(n), with the Fibonacci numbers F(n) = A000045(n).
Sum_{k=0..n} T(n, k) = (2^(2*n + 1) + 1)/3 = A007583(n).
G.f. for column m >= 0: x^m*pFe(m+1, x)/(1-3*x+x^2)^(m+1), where pFe(n, x) := Sum_{m=0..n} A061176(n, m)*x^m (row polynomials of signed triangle A061176).
G.f.: (1-x*(1+y))/(1 - (3+2*y)*x + (1+y)^2*x^2). - Vladeta Jovovic, Oct 11 2003

A060921 Bisection of Fibonacci triangle A037027: odd-indexed members of column sequences of A037027 (not counting leading zeros).

Original entry on oeis.org

1, 3, 2, 8, 10, 3, 21, 38, 22, 4, 55, 130, 111, 40, 5, 144, 420, 474, 256, 65, 6, 377, 1308, 1836, 1324, 511, 98, 7, 987, 3970, 6666, 6020, 3130, 924, 140, 8, 2584, 11822, 23109, 25088, 16435, 6588, 1554, 192, 9

Offset: 0

Wolfdieter Lang, Apr 20 2001

Row sums give A002450. Column sequences (without leading zeros) give for m=0..5: A001906, 2*A001870, A061182, 4*A061183, A061184, 2*A061185.

Companion triangle (odd-indexed members) A060920.

			{1}; {3,2}; {8,10,3}; {21,38,22,4}; ...; pFo(2,x) = 2*(1-x).

Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, Nov. 2005, pp. 359-370.

a(n, m) = A037027(2*n+1-m, m).
a(n, m) = (2*(n-m+1)*A060920(n, m-1)+2*(2*n+1)*a(n-1, m-1))/(5*m), n >= m>0; a(n, 0) := S(n, 3)=A001906(n+1) with Chebyshev's S(n, x) polynomials A049310; else 0.
G.f. for column m >= 0: x^m*pFo(m+1, x)/(1-3*x+x^2)^(m+1), where pFo(n, x) := Sum_{m=0..n-1} A061177(n-1, m)*x^m (row polynomials of signed triangle A061177).
G.f.: 1/(1 - (3+2*y)*x + (1+y)^2*x^2). - Vladeta Jovovic, Oct 11 2003

A038112 a(n) = T(2n,n), where T(n,k) is in A037027.

Original entry on oeis.org

1, 2, 9, 40, 190, 924, 4578, 22968, 116325, 593450, 3045185, 15699840, 81260816, 421993040, 2197653240, 11472991008, 60023749566, 314621200260, 1651883008050, 8685998428800, 45734484854520, 241098942106440, 1272406536645660

Offset: 0

Floor van Lamoen

Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,0). - Joerg Arndt, Jun 30 2011

Diagonal of rational function 1/(1 - (x + y + y^2)). - Gheorghe Coserea, Aug 06 2018

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 40*x^3 + 190*x^4 + 924*x^5 + 4578*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
P. Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3 example 7.

List([0..25],n->Sum([0..n],k->Binomial(n+k,k)*Binomial(k,n-k))); # Muniru A Asiru, Aug 06 2018

a:=n->sum(binomial(2*j,n)*(binomial(n+j,2*j)),j=0..n): seq(a(n), n=0..21); # Zerinvary Lajos, Aug 22 2006
series(RootOf((x+1)*(27*x-5)*A^3+4*A+1,A),x=0,30); # Mark van Hoeij, May 01 2013

Table[Sum[Binomial[n+k,k]Binomial[k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Sep 30 2012 *)
Table[Binomial[2 n, n] Hypergeometric2F1[1/2 - n/2, -n/2, -2 n, -4], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 19 2016 *)

{a(n) = if( n<0, 0, sum(k=0, n\2, (2*n-k)!/ (k! * (n-2*k)!)) / n!)}; /* Michael Somos, Sep 29 2003 */

{a(n) = if( n<0, 0, n++; n * polcoeff(serreverse( x - x^2 - x^3 + x * O(x^n)), n))}; /* Michael Somos, Sep 29 2003 */

/* same as in A092566 but use */
steps=[[0,1], [1,0], [2,0]]; /* Joerg Arndt, Jun 30 2011 */

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ = d^n/dx^n F
{a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, x^(2*m)*(1+x+x*O(x^n))^m/m!)); polcoeff(A, n)} \\ Paul D. Hanna, Aug 04 2012

a(n) = Sum_{k=0..n} C(n+k,k)*C(k,n-k). - Paul Barry, May 13 2006
a(n) = Sum_{j=0..n} binomial(2*j, n)*binomial(n+j, 2*j). - Zerinvary Lajos, Aug 22 2006
a(n) = [x^n] (1/(1-x-x^2))^(n+1). - Paul Barry, Mar 23 2011
a(n) = (n+1)*A001002(n+1).
G.f.: Sum_{n>=0} d^n/dx^n x^(2*n)*(1+x)^n/n!. - Paul D. Hanna, Aug 04 2012
Recurrence: 5*(n-1)*n*a(n) = 11*(n-1)*(2*n-1)*a(n-1) + 3*(3*n-4)*(3*n-2)*a(n-2). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 3^(3*n+3/2)/(2^(3/2)*5^(n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
G.f.: A(x) where (x+1)*(27*x-5)*A(x)^3 + 4*A(x) + 1 = 0. - Mark van Hoeij, May 01 2013

A038149 a(n) = max T(n,k), with T as in A037027.

Original entry on oeis.org

1, 1, 2, 5, 10, 22, 51, 111, 256, 594, 1324, 3130, 7285, 16435, 39430, 91959, 211519, 506408, 1182650, 2760810, 6592080, 15409740, 36369225, 86666790, 202738635, 482548825, 1148125550, 2687285975, 6439167735, 15301850850, 35843163720

Offset: 0

Floor van Lamoen

nonn

Cf. A037027.

A054446 Triangle of partial row sums of triangle A037027(n,m), n >= m >= 0 (Fibonacci convolution triangle).

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 12, 9, 4, 1, 29, 24, 14, 5, 1, 70, 62, 42, 20, 6, 1, 169, 156, 118, 67, 27, 7, 1, 408, 387, 316, 205, 100, 35, 8, 1, 985, 951, 821, 588, 332, 142, 44, 9, 1, 2378, 2323, 2088, 1614, 1020, 509, 194, 54, 10, 1, 5741, 5652, 5232, 4290, 2966, 1671, 747, 257

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is Pell(z)/(1-x*z*Fib(z)) with Pell(x)=1/(1-2*x-x^2) = g.f. for A000129(n+1) (Pell numbers without 0) and Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).

			{1}; {2,1}; {5,3,1}; {12,9,4,1};...
Fourth row polynomial (n=3): p(3,x)= 12+9*x+4*x^2+x^3

Cf. A037027, A000045, A000129. Row sums: A054447(n).

a(n, m)=sum(A037027(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j-1, m)*A037027(n-j, 0), j=m..n) + A037027(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: Pell(x)*(x*Fib(x))^m, m >= 0, with Fib(x) = g.f. A000045(n+1) and Pell(x) = g.f. A000129(n+1).

T(n,0) = 2*T(n-1,0) + T(n-2,0), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k) for k>0, T(0,0) = 1, T(1,0) = 2, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 26 2014

A181347 Numerators of lower triangular matrix T:=log(F), with the matrix F:=A037027 (Fibonacci convolution matrix).

Original entry on oeis.org

0, 1, 0, 1, 2, 0, -1, 2, 3, 0, -1, -1, 3, 4, 0, 2, -2, -3, 4, 5, 0, 2, 4, -1, -2, 5, 6, 0, -31, 4, 2, -4, -5, 6, 7, 0, -3, -31, 2, 8, -5, -3, 7, 8, 0, 202, -3, -31, 8, 10, -2, -7, 8, 9, 0, 4, 404, -9, -62, 2, 4, -7, -4, 9, 10, 0, -464, 8, 202, -6, -31, 4, 14, -8, -9, 10, 11, 0, -2048, -928, 12, 808, -3, -31, 14, 16, -3, -5, 11, 12, 0

Offset: 0

Wolfdieter Lang, Oct 15 2010

The denominator triangle is given by A181348.
Because exp(T) = F, T may be considered as generator of F.
This should be read as N x N matrix for N>=2: log(F_N) := -sum(((-1)^k)/k)*(F_N - Id_N)^k,N-1) with the lower triangular N x N matrix F_N := Matrix([seq([seq(A037027(n,m),n=0..N-1)],m=0..N-1)]) and the N x N identity matrix Id_N.
The log series terminates because of the lower triangular property, and the fact that all main diagonal elements are 1, which follows from
F = Riordan matrix (Fib(x),x*Fib(x)) with the o.g.f. Fib(x)=1/(1-x-x^2). For this notation of Riordan arrays see, e.g., the W.Lang link given in A006232, and there the paragraph "Summary on A- and Z-sequences for Riordan matrices", as well as the 1991 Shapiro et al. reference on the Riordan group given in A053121.

			[0]; [1,0]; [1,2,0]; [-1,2,3,0]; [-1,-1,3,4,0];...
The rational triangle (with main diagonal elements 0) starts with the rows [0]; [1, 0]; [1, 2, 0]; [-1/2, 2, 3, 0]; [-1/3, -1, 3, 4, 0];...

Wolfdieter Lang, A181347/A181348, Logarithm of the Fibonacci matrix A037027

a(n,m) = numerator((log F)(n,m)), with the Fibonacci lower triangular matrix F=A037027.

A181348 Denominators of lower triangular matrix T:=log(F), with the matrix F:=A037027 (Fibonacci convolution matrix).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 2, 1, 1, 1, 15, 3, 1, 1, 1, 1, 1, 30, 15, 1, 3, 2, 1, 1, 1, 70, 15, 5, 3, 3, 1, 1, 1, 1, 105, 35, 10, 15, 3, 1, 2, 1, 1, 1, 7, 105, 70, 15, 3, 1, 3, 1, 1, 1, 1, 105, 7, 35, 35, 6, 5, 3, 3, 2, 1, 1, 1, 385, 105, 7, 105, 14, 5, 15, 3, 1, 1, 1, 1, 1, 3465

Offset: 0

Wolfdieter Lang, Oct 15 2010

The triangle of the numerators is given by A181347.

See the comments given in A181347.

			The triangle starts with the rows[1]; [1,1]; [1,1,1]; [2,1,1,1]; [3,1,1,1,1]; ...
The rational triangle T (with main diagonal elements 0) starts with the rows [0]; [1, 0]; [1, 2, 0]; [-1/2, 2, 3, 0]; [-1/3, -1, 3, 4, 0];...

Wolfdieter Lang, A181347/A181348, Logarithm of the Fibonacci matrix A037027

Cf. A037027, A181347.

a(n,m) = denominator((log F)(n,m)), with the Fibonacci lower triangular matrix F=A037027.

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A055830 Triangle T read by rows: diagonal differences of triangle A037027.

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A111006 Another version of Fibonacci-Pascal triangle A037027.

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A038137 Reflection of A037027: T(n,m) = U(n,n-m), m=0..n, where U is as in A037027.

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A060920 Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).

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A060921 Bisection of Fibonacci triangle A037027: odd-indexed members of column sequences of A037027 (not counting leading zeros).

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A038112 a(n) = T(2n,n), where T(n,k) is in A037027.

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A038149 a(n) = max T(n,k), with T as in A037027.

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