A073370 -id:A073370 - cp's OEIS frontend

A073377 Seventh convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

1, 8, 52, 264, 1194, 4872, 18516, 66264, 226083, 740608, 2344232, 7202416, 21562164, 63090288, 180884088, 509245776, 1410356133, 3848340312, 10359516684, 27544099704, 72406891326, 188356187448

Offset: 0

Wolfdieter Lang, Aug 02 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-12,-56,154,168,-700,-328,1791,656,-2800,-1344, 2464,1792,-768,-1024,-256).

Eighth (m=7) column of triangle A073370.

Cf. A001045, A073376.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^8 )); // G. C. Greubel, Sep 29 2022

Table[(2^(n+8)*(9539600 +17240268*n +11555460*n^2 +3849489*n^3 +703080*n^4 +71442*n^5 +3780*n^6 +81*n^7) +(-1)^n*(236325040 +225702732*n +87290028*n^2 +17880849*n^3 +2109240*n^4 +144018*n^5 +5292*n^6 +81*n^7))/(7!*3^12), {n,0,60}] (* G. C. Greubel, Sep 29 2022 *)

def A073377(n): return (2^(n+8)*(9539600 +17240268*n +11555460*n^2 +3849489*n^3 +703080*n^4 +71442*n^5 +3780*n^6 +81*n^7) +(-1)^n*(236325040 +225702732*n +87290028*n^2 +17880849*n^3 +2109240*n^4 +144018*n^5 +5292*n^6 +81*n^7))/(factorial(7)*3^12)
[A073377(n) for n in range(40)] # G. C. Greubel, Sep 29 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073376(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+7, 7) * binomial(n-k, k) * 2^k.
a(n) = ((328247920 +332102604*n +131833680*n^2 +26450901*n^3 +2844099*n^4 + 156087*n^5 +3429*n^6)*(n+1)*U(n+1) + 2(141143240 +150941694*n +62335731*n^2 + 12873492*n^3 +1414314*n^4 +78894*n^5 +1755*n^6)*(n+2)*U(n))/(7!*3^11) with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1-(1+2*x)*x)^8 = 1/((1+x)*(1-2*x))^8.
E.g.f.: (1/(7!*3^12))*( 4096*(596225 +4177950*x +7304850*x^2 +5109300*x^3 +1691550*x^4 +278964*x^5 +21924*x^6 +648*x^7)*exp(2*x) + (236325040 -333132240*x +158026680*x^2 -34637400*x^3 +3921750*x^4 -234738*x^5 +6993*x^6 -81*x^7)*exp(-x) ). - G. C. Greubel, Sep 29 2022

A073378 Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 9, 63, 345, 1665, 7227, 29073, 109791, 394020, 1354210, 4486482, 14397318, 44932446, 136817370, 407566350, 1190446866, 3415935699, 9645169743, 26836557825, 73670997015, 199751003991, 535449185469

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n) with U(n) = A001045(n+1) see A073370 and the row polynomials of triangles A073399 and A073400.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-18,-60,234,126,-1176,36,3519,-479,-7038,144, 9408,2016,-7488,-3840,2304,2304,512).

Ninth (m=8) column of triangle A073370.

Cf. A001045, A073377, A073399, A073400, A073401.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^9 )); // G. C. Greubel, Oct 01 2022

CoefficientList[Series[1/((1+x)*(1-2*x))^9, {x,0,40}], x] (* G. C. Greubel, Oct 01 2022 *)

def A073378_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*(1-2*x))^9 ).list()
A073378_list(40) # G. C. Greubel, Oct 01 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073377(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+8, 8) * binomial(n-k, k) * 2^k.
G.f.: 1/(1-(1+2*x)*x)^9 = 1/((1+x)*(1-2*x))^9.

A073402 Coefficient triangle of polynomials (rising powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073401.

Original entry on oeis.org

2, 33, 9, 831, 396, 45, 28236, 18297, 3744, 243, 1210140, 968679, 273483, 32481, 1377, 62686440, 58920534, 20681811, 3418767, 268029, 8019, 3810867480, 4075425738, 1683064737, 347584284, 38186478, 2130138, 47385

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are q(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of U0(n) := A001045(n+1), n>= 0, ((1,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073370(n+k,k) = (p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*2*U0(n))/(k!*9^k)), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073401(k,m).

			k=2: U2(n)=((30+9*n)*(n+1)*U0(n+1)+(33+9*n)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 33,9; 831,396,45; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Sean A. Irvine, Table of n, a(n) for n = 0..989
Wolfdieter Lang, First 7 rows.

Cf. A001045, A073370, A073399, A073372, A073400.

Recursion for row polynomials defined in the comments: p(k, n)= (n+2)*p(k-1, n+1)+4*(n+2*(k+1))*p(k-1, n)+2*(n+3)*q(k-1, n+1); q(k, n)= (n+1)*p(k-1, n+1)+4*(n+2*(k+1))*q(k-1, n), k >= 1. [Corrected by Sean A. Irvine, Nov 25 2024]

A073379 Ninth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 10, 75, 440, 2255, 10362, 43945, 174460, 656370, 2359500, 8158722, 27275040, 88524930, 279892380, 864508590, 2614740216, 7759693095, 22634343270, 64990287285, 183929970840, 513661549401, 1416970676550

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n), with U(n) = A001045(n+1), see A073370 and the row polynomials of triangles A073399 and A073400.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-25,-60,330,12,-1770,960,5835,-4070,-13597, 8140,23340,-7680,-28320,-384,21120,7680,-6400,-5120,-1024).

Tenth (m=9) column of triangle A073370.

Cf. A001045, A073370, A073378, A073399, A073400.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^10 )); // G. C. Greubel, Oct 01 2022

CoefficientList[Series[1/((1+x)*(1-2*x))^10, {x,0,40}], x] (* G. C. Greubel, Oct 01 2022 *)

def A073379_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*(1-2*x))^10 ).list()
A073379_list(40) # G. C. Greubel, Oct 01 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073378(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+9, 9) * binomial(n-k, k) * 2^k.
G.f.: 1/(1-(1+2*x)*x)^10 = 1/((1+x)*(1-2*x))^10.

A125693 Riordan array ((1-x)/(1-3x), x(1-x)/(1-3*x)).

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 18, 16, 6, 1, 54, 60, 30, 8, 1, 162, 216, 134, 48, 10, 1, 486, 756, 558, 248, 70, 12, 1, 1458, 2592, 2214, 1168, 410, 96, 14, 1, 4374, 8748, 8478, 5160, 2150, 628, 126, 16, 1, 13122, 29160, 31590, 21744, 10442, 3624, 910, 160, 18, 1

Offset: 0

Paul Barry, Nov 30 2006

Row sums are A001835(n+1). Diagonal sums are A030186. Inverse is A125694. Equal to product of A007318 and A073370.

			Triangle begins
    1;
    2,   1;
    6,   4,   1;
   18,  16,   6,  1;
   54,  60,  30,  8,  1;
  162, 216, 134, 48, 10, 1;

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

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j->
(-1)^j*3^(n-k-j)*Binomial(k+1,j)*Binomial(n-j, n-k-j) )))); # G. C. Greubel, Oct 28 2019

T:= func< n,k | &+[(-1)^j*3^(n-k-j)*Binomial(k+1,j)*Binomial(n-j, n-k-j): j in [0..n]] >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 28 2019

seq(seq( add( (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j), j=0..n), k=0..n), n=0..10); # G. C. Greubel, Oct 28 2019

T[0, 0]=1; T[1, 0]=2; T[1, 1]=1; T[n_, k_]/; 0<=k<=n:= T[n, k]= 3T[n-1, k] + T[n-1, k-1] - T[n-2, k-1]; T[, ]=0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)
T[n_, k_]:= Sum[(-1)^j*3^(n-k-j)*Binomial[k+1,j]*Binomial[n-j,n-k-j], {j, 0, n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 28 2019 *)

T(n,k) = sum(j=0,n, (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 28 2019

[[sum( (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j) for j in (0..n) ) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Oct 28 2019

Number triangle T(n,k) = Sum_{j=0..k+1} C(k+1,j)*C(n-j,n-k-j)* (-1)^j * 3^(n-k-j).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(n,k)=0 if k>n or if kPhilippe Deléham, Jan 08 2013

A112883 A skew Jacobsthal-Pascal matrix.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 0, 2, 5, 0, 0, 1, 7, 11, 0, 0, 0, 3, 16, 21, 0, 0, 0, 1, 12, 41, 43, 0, 0, 0, 0, 4, 34, 94, 85, 0, 0, 0, 0, 1, 18, 99, 219, 171, 0, 0, 0, 0, 0, 5, 60, 261, 492, 341, 0, 0, 0, 0, 0, 1, 25, 195, 678, 1101, 683, 0, 0, 0, 0, 0, 0, 6, 95, 576, 1692, 2426, 1365, 0, 0, 0, 0, 0

Offset: 0

Paul Barry, Oct 05 2005

T(n,n) is A001045(n), row sums are A006130, column sums are A002605. Compare with [0,1,-1,0,0,..] DELTA [1,2,-2,0,0,...] where DELTA is the operator defined in A084938. A skewed version of the Riordan array (1/(1-x-2x^2),x/(1-x-2x^2)) (A073370).

Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008

			Rows begin
  1;
  0, 1;
  0, 1, 3;
  0, 0, 2, 5;
  0, 0, 1, 7, 11;
  0, 0, 0, 3, 16, 21;
  0, 0, 0, 1, 12, 41, 43;
  0, 0, 0, 0,  4, 34, 94,  85;
  0, 0, 0, 0,  1, 18, 99, 219, 171;
  0, 0, 0, 0,  0,  5, 60, 261, 492,  341;
  0, 0, 0, 0,  0,  1, 25, 195, 678, 1101, 683;

Cf. A111006.

From Philippe Deléham: (Start)
G.f.: 1/(1-yx(1-x)-2x^2*y*2);
Number triangle T(n, k) = Sum_{j=0..2k-n} C(n-k+j, n-k)*C(j, 2k-n-j)*2^(2k-n-j);
T(n, k) = A073370(k, n-k); T(n, k) = T(n-1, k-1) + T(n-2, k-1) + 2*T(n-2, k-2). (End)

A117316 Riordan array ((1-x)/(1-x-2x^2),x(1-x)/(1-x-2x^2)).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 2, 4, 0, 1, 6, 4, 6, 0, 1, 10, 16, 6, 8, 0, 1, 22, 28, 30, 8, 10, 0, 1, 42, 72, 54, 48, 10, 12, 0, 1, 86, 148, 158, 88, 70, 12, 14, 0, 1, 170, 336, 342, 288, 130, 96, 14, 16, 0, 1, 342, 716, 846, 648, 470, 180, 126, 16, 18, 0, 1

Offset: 0

Paul Barry, Mar 07 2006

Product of A007318 and alternating sign version of A073370. Row sums are A001333. Diagonal sums are A052973. First column is A078008. Convolution array for A078008.

			Triangle begins
1,
0, 1,
2, 0, 1,
2, 4, 0, 1,
6, 4, 6, 0, 1,
10, 16, 6, 8, 0, 1,
22, 28, 30, 8, 10, 0, 1,
42, 72, 54, 48, 10, 12, 0, 1

T(n,k)=sum{j=0..n-k,sum{i=0..k+1, C(k+1,i)C(k+j,j)C(n-i-j,n-k-i-j)(-1)^(i+j)2^(n-k-i-j)}}

T(n,k)=T(n-1,k)+T(n-1,k-1)+2*T(n-2,k)-T(n-2,k-1), T(0,0)= 1, T(1,0)=0, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 04 2013

A179533 Expansion of (1/(1-x-2x^2))*c(x/(1-x-2x^2)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 7, 23, 85, 332, 1369, 5870, 25945, 117374, 540805, 2528675, 11966923, 57206972, 275824159, 1339721519, 6549093013, 32195473406, 159065828029, 789395034701, 3933239089903, 19668745466636, 98679891233803, 496570499905832, 2505670304785615, 12675395921692394, 64270076976110203, 326580624341708693, 1662796531746045157, 8481930651824392268, 43341418581113085697

Offset: 0

Paul Barry, Jan 08 2011

nonn

Hankel transform is A168495(n+1).

Cf. A000108, A073370, A168495.

with(LREtools): with(FormalPowerSeries): # requires Maple 2022
ogf:= (1/(1-x-2*x^2))*(1 - sqrt(1 - 4*(x/(1-x-2*x^2)))) / (2*(x/(1-x-2*x^2))):
init:= [1, 2, 7, 23, 85, 332, 1369];
iseq:= seq(u(i-1)=init[i],i=1..nops(init)): req:= FindRE(ogf,x,u(n));
rmin:= subs(n=n-4,MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
seq(a(n),n=0..30); # Georg Fischer, Nov 04 2022

G.f.: 1/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-... (continued fraction);
a(n) = Sum_{k=0..n} A000108(k)*Sum_{j=0..n-k} C(k+j,k)*C(j,n-k-j)*2^(n-k-j).
a(n) = Sum_{k=0..n} A073370(n,k)*A000108(k).
D-finite with recurrence: (n+1)*a(n) +2*(1-3n)*a(n-1) +(n-1)*a(n-2) +4*(3n-5)*a(n-3) +4*(n-3)*a(n-4)= 0. - R. J. Mathar, Nov 17 2011

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A073377 Seventh convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

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A073378 Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

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A073402 Coefficient triangle of polynomials (rising powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073401.

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A073379 Ninth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

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A125693 Riordan array ((1-x)/(1-3*x), x*(1-x)/(1-3*x)).

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A112883 A skew Jacobsthal-Pascal matrix.

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A117316 Riordan array ((1-x)/(1-x-2x^2),x(1-x)/(1-x-2x^2)).

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A179533 Expansion of (1/(1-x-2x^2))*c(x/(1-x-2x^2)), c(x) the g.f. of A000108.

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A125693 Riordan array ((1-x)/(1-3x), x(1-x)/(1-3*x)).