A004799 -id:A004799 - cp's OEIS frontend

A067979 Triangle read by rows of incomplete convolutions of Lucas numbers L(n+1) = A000204(n+1), n>=0.

1, 3, 6, 4, 13, 17, 7, 19, 31, 38, 11, 32, 48, 69, 80, 18, 51, 79, 107, 140, 158, 29, 83, 127, 176, 220, 274, 303, 47, 134, 206, 283, 360, 432, 519, 566, 76, 217, 333, 459, 580, 706, 822, 963, 1039, 123, 351, 539, 742, 940, 1138, 1341, 1529, 1757, 1880, 199, 568, 872, 1201, 1520, 1844, 2163, 2492, 2796, 3165, 3364

Offset: 0

Wolfdieter Lang, Feb 15 2002

The diagonals d>=0 (d=0: main diagonal) give convolutions of Lucas numbers L(n+1) := A000204(n+1), n>=0, with those with d-shifted index: a(d+n,d) = Sum_{k=0..n} L(k+1)*L(d+n+1-k).
The diagonals give A004799(n-1), A067980-7 for d=n-m= 0..8, respectively. Row sums give A067989.
The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(x*z)*(A(z)-x*A(x*z))/(1-x), with A(x) := (1+2*x)/(1-x-x^2) (g.f. Lucas L(n+1), n>=0).

			Triangle begins:
  {1};
  {3,6};
  {4,13,17};     p(2,x) = 4+13*x+17*x^2
  {7,19,31,38};
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A067990 (triangle with rows read backwards).

Table[Sum[LucasL[k + 1] LucasL[n - k + 1], {k, 0, m}], {n, 0, 10}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

for(n=0,10, for(k=0,n, print1(sum(m=0,k,(fibonacci(m+2) + fibonacci(m))*(fibonacci(n-m+2) + fibonacci(n-m))), ", "))) \\ G. C. Greubel, Dec 17 2017

a(n, m) = Sum_{k=0..m} L(k+1)*L(n-k+1), n>=m>=0, else 0.
a(n, m) = (m+1)*L(n-m+1)*F(m) + ((m+1)*L(n-m+1) + m*L(n-m))*F(m+1), n>=m>=0, with F(n) := A000045(n) (Fibonacci) and L(n) := A000032(n) (Lucas).
G.f. for diagonals d= n-m>=0: (x^d)*(L(d+1)+L(d)*x)*(1-2*x)/(1-x-x^2)^2.
a(n, m) = -(-1)^m*F(n-2*m-1) + m*L(n+2)+F(n+3), with F(-n) = (-1)^(n+1) * F(n), hence a(n, m) = -5*A067330(n, m)+2*(m+1)*L(n+2), n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A067990 Triangle A067979 with rows read backwards.

Original entry on oeis.org

1, 6, 3, 17, 13, 4, 38, 31, 19, 7, 80, 69, 48, 32, 11, 158, 140, 107, 79, 51, 18, 303, 274, 220, 176, 127, 83, 29, 566, 519, 432, 360, 283, 206, 134, 47, 1039, 963, 822, 706, 580, 459, 333, 217, 76, 1880, 1757, 1529, 1341, 1138, 940, 742, 539, 351, 123, 3364, 3165, 2796, 2492, 2163, 1844, 1520, 1201

Offset: 0

Wolfdieter Lang, Feb 15 2002

The column m (without leading 0's) gives the convolution of Lucas numbers {L(n+1) := A000204(n+1)}, n>=0, with those with m-shifted index: a(n+m,m)=sum(L(k+1)*L(m+n+1-k),k=0..n), n>=0,m=0,1,...
The columns give A004799(n-1), A067980-7 for m= 0..8, respectively. Row sums give A067989.
The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(z)*(A(z)-x*A(x*z))/(1-x), with A(x) := (1+2*x)/(1-x-x^2) (g.f. for Lucas {L(n+1)}).

			{1}; {6,3}; {17,13,4}; {38,31,19,7}; ...; p(2,x)=17+13*x+4*x^2.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Reverse /@ Table[Sum[LucasL[k + 1] LucasL[n - k + 1], {k, 0, m}], {n, 0, 11}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

a(n, m)=A067330(n, n-m), n>=m>=0, else 0.
a(n, m)=(n-m+1)*L(m+1)*F(n-m)+((n-m+1)*L(m+1)+(n-m)*L(m))*F(n-m+1), n>=m>=0, else 0; with F(n) := A000045(n)(Fibonacci) and L(n) := A000032(n) (Lucas).
G.f. for column m=0, 1, ...: (x^m)*(L(m+1)+L(m)*x)*(1+2*x)/(1-x-x^2)^2.
a(n, m) = -(-1)^m*F(n-2*m+1)-m*L(n+2)+n*L(n+2)+F(n+3), with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = -5*A067418(n, m)+2*(n-m+1)*L(n+2), n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A060922 Convolution triangle for Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 3, 1, 4, 6, 1, 7, 17, 9, 1, 11, 38, 39, 12, 1, 18, 80, 120, 70, 15, 1, 29, 158, 315, 280, 110, 18, 1, 47, 303, 753, 905, 545, 159, 21, 1, 76, 566, 1687, 2568, 2120, 942, 217, 24, 1, 123, 1039, 3612, 6666, 7043, 4311

Offset: 0

Wolfdieter Lang, Apr 20 2001

In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. G.f. for row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) is (1+2*z)/(1-(1+x)*z-(1+2*x)*z^2).
Row sums give A060925. Column sequences (without leading zeros) are, for m=0..6: A000032(n+1)= A000204(n+1) (Lucas), A004799(n+1), A060929-33.
Bisection of this triangle gives triangles A060923 (even part) and A060924 (odd part).
For the m-th column sequence (without leading zeros) one has: a(n+m,m)= (pL1(m,n)*L(n+2)+pL2(m,n)*L(n+1))/(m!*5^m), m >= 0, with the Lucas numbers L(n)=A000032(n), n >= 0 and the row polynomials pL1(n,x) := sum(A061188(n,m)*x^n,m=0..n) and pL2(n,x) := sum(A061189(n,m)*x^m,m=0..n).
Riordan array ((1+2*x)/(1-x-x^2), x*(1+2*x)/(1-x-x^2)). - Philippe Deléham, Jan 21 2014
T is the convolution triangle of A000204 (see A357368). - Peter Luschny, Oct 19 2022

			p(2,x) = 4+6*x+x^2.
Triangle begins:
1 ;
3, 1;
4, 6, 1;
7, 17, 9, 1;
11, 38, 39, 12, 1;
18, 80, 120, 70, 15, 1;
29, 158, 315, 280, 110, 18, 1;
47, 303, 753, 905, 545, 159, 21, 1;

Cf. A000032.

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, A000204); # Peter Luschny, Oct 19 2022

a(n, m)=((n-m+1)*a(n, m-1)+2*(2*n-m)*a(n-1, m-1)+4*(n-1)*a(n-2, m-1))/(5*m), n >= m >= 1, a(n, 0)= A000204(n+1)= A000032(n+1).
G.f. for m-th column: ((1+2*x)/(1-x-x^2))* ((x*(1+2*x))/(1-x-x^2))^m.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 21 2014

Example improved by Philippe Deléham, Jan 21 2014

A060929 Second convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 9, 39, 120, 315, 753, 1687, 3612, 7470, 15040, 29634, 57366, 109421, 206115, 384105, 709152, 1298613, 2360943, 4264835, 7659870, 13686456, 24340184, 43102644, 76031100, 133636825, 234116493, 408900987

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A004799, A060922.

I:=[1,9,39,120,315,753]; [n le 6 select I[n] else 3*Self(n-1) - 5*Self(n-3) + 3*Self(n-5) + Self(n-6): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CoefficientList[Series[((1 + 2*x)/(1 - x - x^2))^3, {x, 0, 50}], x] (* or *) LinearRecurrence[{3,0,-5,0,3,1}, {1,9,39,120,315,753}, 30] (* G. C. Greubel, Dec 21 2017 *)

x='x+O('x^30); Vec(((1+2*x)/(1-x-x^2))^3) \\ G. C. Greubel, Dec 21 2017

G.f.: ((1+2*x)/(1-x-x^2))^3.
a(n) = A060922(n+2, 2) (third column of Lucas triangle).
a(n) = (n+1)*((5*n+4)*L(n+2) + (5*n-2)*L(n+1))/10, n >= 1, with the Lucas numbers L(n)=A000032(n)=A000204(n), n >= 1.

A060930 Third convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 12, 70, 280, 905, 2568, 6666, 16220, 37580, 83780, 181074, 381488, 786715, 1593160, 3176210, 6246732, 12139859, 23344760, 44471340, 84005640, 157483176, 293201912, 542468100, 997906400, 1826073525

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,5,8,-2,-4,-1).

Cf. A000032, A000204, A004799, A060922, A060929.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^4 )); // G. C. Greubel, Apr 08 2021

Table[((25*n^3+90*n^2+95*n+6)*LucasL[n+4] -12*(5*n^2+10*n-3)*LucasL[n+2])/150, {n, 0, 40}] (* G. C. Greubel, Apr 08 2021 *)

def A060930_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^4 ).list()
A060930_list(40) # G. C. Greubel, Apr 08 2021

G.f.: ((1+2*x)/(1-x-x^2))^4.
a(n) = A060922(n+3, 3) (fourth column of Lucas triangle).
a(n) = (2*(25*n^3 + 60*n^2 + 35*n +24)*L(n+2) + (25*n^3 + 90*n^2 + 95*n + 6)*L(n+1))/(3!*5^2), with the Lucas numbers L(n) = A000032(n).

A060931 Fourth convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 15, 110, 545, 2120, 7043, 20965, 57560, 148545, 365045, 862224, 1970905, 4382820, 9520315, 20265665, 42385132, 87284120, 177293730, 355738710, 705980760, 1387213926, 2701362950, 5217448800, 10001654350

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-5,-10,15,11,-15,-10,5,5,1).

Cf. A000032, A000204, A004799, A060922, A060929, A060930.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^5 )); // G. C. Greubel, Apr 08 2021

Table[((n+1)/120)*((5*n^3+5*n^2-10*n+72)*LucasL[n+5] + 4*(5*n^2+10*n-24)*LucasL[n+ 4]), {n, 0, 40}] (* G. C. Greubel, Apr 08 2021 *)

def A060931_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^5 ).list()
A060931_list(40) # G. C. Greubel, Apr 08 2021

a(n) = A060921(n+4, 4) (fifth column of Lucas triangle).
a(n) = (n+1)*( (15*n^3 +55*n^2 +50*n +24)*L(n+2) + 2*(5*n^3 +15*n^2 +10*n +24)*L(n+1))/5!, with the Lucas numbers L(n)=A000032(n).
G.f.: ((1+2*x)/(1-x-x^2))^5.

A060932 Fifth convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 18, 159, 942, 4311, 16536, 55898, 171924, 491487, 1325546, 3409347, 8430246, 20164223, 46880424, 106350942, 236147828, 514553154, 1102562952, 2327442276, 4847463408, 9974081130, 20297335340

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9,-10,30,6,-41,-6,30,10,-9,-6,-1).

Cf. A000032, A000204, A004799, A060922, A060929, A060930, A060931.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^6 )); // G. C. Greubel, Apr 08 2021

Table[((744+2990*n+2895*n^2+1925*n^3+825*n^4+125*n^5)*LucasL[n+2] +3*(256+390*n + 505*n^2+425*n^3+175*n^4+25*n^5)*LucasL[n+1])/(5^2*5!), {n,0,40}] (* G. C. Greubel, Apr 08 2021 *)

def A060932_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^6 ).list()
A060932_list(40) # G. C. Greubel, Apr 08 2021

a(n) = A060922(n+5, 5) (sixth column of Lucas triangle).
G.f.: ((1+2*x)/(1-x-x^2))^6.
a(n) = ( 25*(125*n^5 +825*n^4 +1925*n^3 +2895*n^2 +2990*n +744)*L(n+2) +(1875*n^5 +13125*n^4 +31875*n^3 +37875*n^2 +29250*n +19200)*L(n+1))/(5!*5^4), with the Lucas numbers L(n)=A000032(n).

A060933 Sixth convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 21, 217, 1498, 7910, 34566, 131446, 449732, 1416513, 4174765, 11651717, 31075422, 79751854, 198036146, 477899790, 1124785648, 2589534248, 5845989156, 12968091584, 28316428700, 60953528230, 129515454530, 271955244610, 564879359940, 1161646929275, 2366938010983, 4781794056543

Offset: 0

Wolfdieter Lang, Apr 20 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,-7,49,-14,-77,29,77,-14,-49,-7,14,7,1).

Cf. A000032, A000204, A004799, A060922, A060929, A060930, A060931, A060932.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^7 )); // G. C. Greubel, Apr 08 2021

m:= 40; S:= series( ((1+2*x)/(1-x-x^2))^7, x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Apr 08 2021

Table[(n+1)(2(100n^5+845n^4+2480n^3+4345n^2+5910n+2952)LucasL[n+2]+(125n^5+ 1030n^4+2995n^3+5930n^2+8280n+288)LucasL[n+1])/18000,{n,0,30}] (* Harvey P. Dale, Aug 13 2013 *)
CoefficientList[Series[((1+2x)/(1-x-x^2))^7, {x,0,30}], x] (* Vincenzo Librandi, Aug 13 2013 *)

def A060930_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^7 ).list()
A060930_list(40) # G. C. Greubel, Apr 08 2021

G.f.: ( (1+2*x) / (1-x-x^2) )^7.
a(n) = A060922(n+6, 6) (seventh column of Lucas triangle).
a(n) = (n+1)*(2*(100*n^5 +845*n^4 +2480*n^3 +4345*n^2 +5910*n +2952)*L(n+2) + (125*n^5 +1030*n^4 +2995*n^3 +5930*n^2 +8280*n +288)*L(n+1))/(6!*5^2), with the Lucas numbers L(n)=A000032(n).

cp's OEIS Frontend

A067979 Triangle read by rows of incomplete convolutions of Lucas numbers L(n+1) = A000204(n+1), n>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A067990 Triangle A067979 with rows read backwards.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

Formula

A060922 Convolution triangle for Lucas numbers A000032(n+1), n >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A060929 Second convolution of Lucas numbers A000032(n+1), n >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A060930 Third convolution of Lucas numbers A000032(n+1), n >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A060931 Fourth convolution of Lucas numbers A000032(n+1), n >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A060932 Fifth convolution of Lucas numbers A000032(n+1), n >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A060933 Sixth convolution of Lucas numbers A000032(n+1), n >= 0.

Views