A073399 -id:A073399 - cp's OEIS frontend

A073400 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073399.

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

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are q(k,x) := sum(a(k,m)*x^(k-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^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A073399(k,m).

			k=2: U2(n)=((9*n+30)*(n+1)*U0(n+1)+(9*n+33)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 9,33; 45,396,831; ... (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, A073401.

Recursion for row polynomials defined in the comments: see A073401.

A073370 Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 5, 7, 3, 1, 11, 16, 12, 4, 1, 21, 41, 34, 18, 5, 1, 43, 94, 99, 60, 25, 6, 1, 85, 219, 261, 195, 95, 33, 7, 1, 171, 492, 678, 576, 340, 140, 42, 8, 1, 341, 1101, 1692, 1644, 1106, 546, 196, 52, 9, 1

Offset: 0

Wolfdieter Lang, Aug 02 2002

The g.f. for the row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(1-(1+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.
Riordan array (1/(1-x-2*x^2), x/(1-x-2*x^2)). - Paul Barry, Mar 15 2005
Subtriangle (obtained by dropping the first column) of the triangle given by (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 19 2013
The number of ternary words of length n having k letters equal 2 and 0,1 avoid runs of odd lengths. - Milan Janjic, Jan 14 2017

			Triangle begins as:
    1;
    1,   1;
    3,   2,   1;
    5,   7,   3,   1;
   11,  16,  12,   4,   1;
   21,  41,  34,  18,   5,   1;
   43,  94,  99,  60,  25,   6,   1;
   85, 219, 261, 195,  95,  33,   7,   1;
  171, 492, 678, 576, 340, 140,  42,   8,   1;
The triangle (0, 1, 2, -2, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  3,  2,  1;
  0,  5,  7,  3,  1;
  0, 11, 16, 12,  4,  1;
  0, 21, 41, 34, 18,  5,  1; - _Philippe Deléham_, Feb 19 2013

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
W. Lang, First 10 rows.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Columns: A001045 (k=0), A073371 (k=1), A073372 (k=2), A073373 (k=3), A073374 (k=4), A073375 (k=5), A073376 (k=6), A073377 (k=7), A073378 (k=8), A073379 (k=9).
Cf. A002605 (row sums), A006130 (diagonal sums), A073399, A073400.
Cf. A000045, A077957.

A073370:= func< n,k | (&+[Binomial(n-j,k)*Binomial(n-k-j,j)*2^j: j in [0..Floor((n-k)/2)]]) >;
[A073370(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 01 2022

# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> (2^n - (-1)^n) / 3); # Peter Luschny, Oct 07 2022

T[n_, k_]:= T[n, k]= Sum[Binomial[n-j,k]*Binomial[n-k-j,j]*2^j, {j,0,Floor[(n- k)/2]}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 01 2022 *)

def A073370(n,k): return binomial(n,k)*sum( 2^j * binomial(2*j,j) * binomial(n-k,2*j)/binomial(n,j) for j in range(1+(n-k)//2))
flatten([[A073370(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 01 2022

T(n, m) = Sum_{k=0..floor((n-m)/2)} binomial(n-k, m)*binomial(n-m-k, k)*2^k, if n > m, else 0.
Sum_{k=0..n} T(n, k) = A002605(n+1).
T(n, m) = (1*(n-m+1)*T(n, m-1) + 2*2*(n+m)*T(n-1, m-1))/((1^2 + 4*2)*m), n >= m >= 1, T(n, 0) = A001045(n+1), n >= 0, else 0.
T(n, m) = (p(m-1, n-m)*1*(n-m+1)*T(n-m+1) + q(m-1, n-m)*2*(n-m+2)*T(n-m))/(m!*9^m), n >= m >= 1, with T(n) = T(n, m=0) = A001045(n+1), else 0; p(k, n) = Sum_{j=0..k} (A(k, j)*n^(k-j) and q(k, n) = Sum_{j=0..k} B(k, j)*n^(k-j), with the number triangles A(k, m) = A073399(k, m) and B(k, m) = A073400(k, m).
G.f.: 1/(1-(1+2*x)*x)^(m+1) = 1/((1+x)*(1-2*x))^(m+1), m >= 0, for column m (without leading zeros).
T(n, 0) = A001045(n), T(1, 1) = 1, T(n, k) = 0 if k>n, T(n, k) = T(n-1, k-1) + 2*T(n-2, k) + T(n-1, k) otherwise. - Paul Barry, Mar 15 2005
G.f.: (1+x)*(1-2*x)/(1-x-2*x^2-x*y) for the triangle including the 1, 0, 0, 0, 0, ... column. - R. J. Mathar, Aug 11 2015
From Peter Bala, Oct 07 2019: (Start)
Recurrence for row polynomials: R(n,x) = (1 + x)*R(n-1,x) + 2*R(n-2,x) with R(0,x) = 1 and R(1,x) = 1 + x.
The row reverse polynomial x^n*R(n,1/x) is equal to the numerator polynomial of the finite continued fraction 1 + x/(1 - 2*x/(1 + ... + x/(1 - 2*x/(1)))) (with 2*n partial numerators). Cf. A110441. (End)
From G. C. Greubel, Oct 01 2022: (Start)
T(n, k) = binomial(n,k)*Sum_{j=0..floor((n-k)/2)} 2^j*binomial(2*j, j)*binomial(n-k, 2*j)/binomial(n, j).
T(n, k) = binomial(n, k)*Hypergeometric2F1([(k-n)/2, (k-n+1)/2], [-2*n], -8).
Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006130(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000045(n+1). (End)

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

Original entry on oeis.org

1, 30, 9, 1050, 531, 63, 44520, 29610, 6165, 405, 2245320, 1789614, 502821, 59454, 2511, 131891760, 120133692, 41182344, 6686631, 517104, 15309, 8862693840, 8966770308, 3559509360, 714174327, 76790673, 4214349, 92583

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are p(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 q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073402(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; 30,9; 1050,531,63; ... (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, A073402, 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]

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.

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.

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A073400 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073399.

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A073370 Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.

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

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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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