A074084 -id:A074084 - cp's OEIS frontend

A074082 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,1).

0, 0, 0, 0, 2, 6, 16, 37, 81, 169, 342, 675, 1307, 2491, 4686, 8718, 16066, 29364, 53282, 96065, 172215, 307151, 545286, 963993, 1697701, 2979383, 5211852, 9090060, 15810530, 27429426, 47473828, 81983773, 141286221, 243011173

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

The coefficient of q^0 in nu(n) is the Fibonacci number F(n+1). The coefficient of q^1 is A023610(n-3).

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=2, nu(3)=3+q, nu(4)=5+3q+2q^2, nu(5)=8+7q+6q^2+4q^3+q^4, so the coefficients of q^2 are 0,0,0,0,2,6.

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Coefficients of q^0, q^1 and q^3 are in A000045, A023610 and A074083. Related sequences with different values of b and lambda are in A074084-A074089.

b=1; lambda=1; expon=2; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0, 0}, LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 2, 6, 16, 37}, 32]] (* Jean-François Alcover, Sep 23 2017 *)

G.f.: (2*x^4-2*x^6-x^7)/(1-x-x^2)^3.

a(n) = 3*a(n-1)-5*a(n-3)+3*a(n-5)+a(n-6) for n>=8.

Edited by Dean Hickerson, Aug 21 2002

A074086 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,1).

Original entry on oeis.org

0, 0, 0, 0, 0, 14, 71, 282, 997, 3298, 10439, 32012, 95834, 281494, 814131, 2324422, 6564135, 18362810, 50947395, 140329400, 384031508, 1044880222, 2828084399, 7618214354, 20432838121, 54585196818, 145287466799, 385397215108

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

nonn

The coefficient of q^0 is the Pell number A000129(n+1).

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=5, nu(3)=12+2q, nu(4)=29+9q+5q^2, nu(5)=70+32q+24q^2+14q^3+2q^4, so the coefficients of q^3 are 0,0,0,0,0,14.

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Index entries for linear recurrences with constant coefficients, signature (8, -20, 8, 26, -8, -20, -8, -1).

Coefficients of q^0, q^1 and q^2 are in A000129, A074084 and A074085. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.

b=2; lambda=1; expon=3; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0, 0, 0}, LinearRecurrence[{8, -20, 8, 26, -8, -20, -8, -1}, {0, 0, 14, 71, 282, 997, 3298, 10439}, 25]] (* Jean-François Alcover, Jan 27 2019 *)

G.f.: (14x^5-41x^6-6x^7+49x^8+30x^9+5x^10)/(1-2x-x^2)^4.

a(n) = 8a(n-1)-20a(n-2)+8a(n-3)+26a(n-4)-8a(n-5)-20a(n-6)-8a(n-7)-a(n-8) for n>=11.

Edited by Dean Hickerson, Aug 21 2002

A074083 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,1).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 14, 39, 97, 224, 494, 1051, 2177, 4412, 8784, 17228, 33360, 63886, 121164, 227833, 425147, 787916, 1451198, 2657821, 4842727, 8782230, 15857426, 28517864, 51095760, 91232520, 162372682, 288115147, 509790277, 899630376

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

nonn

The coefficient of q^0 in nu(n) is the Fibonacci number F(n+1). The coefficient of q^1 is A023610(n-3).

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=2, nu(3)=3+q, nu(4)=5+3q+2q^2, nu(5)=8+7q+6q^2+4q^3+q^4, so the coefficients of q^3 are 0,0,0,0,0,4.

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Index entries for linear recurrences with constant coefficients, signature (4, -2, -8, 5, 8, -2, -4, -1).

Coefficients of q^0, q^1 and q^2 are in A000045, A023610 and A074082. Related sequences with different values of b and lambda are in A074084-A074089.

b=1; lambda=1; expon=3; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0, 0, 0}, LinearRecurrence[{4, -2, -8, 5, 8, -2, -4, -1}, {0, 0, 4, 14, 39, 97, 224, 494}, 31]] (* Jean-François Alcover, Jan 27 2019 *)

G.f.: (4x^5-2x^6-9x^7+x^8+6x^9+2x^10)/(1-x-x^2)^4.

a(n) = 4a(n-1)-2a(n-2)-8a(n-3)+5a(n-4)+8a(n-5)-2a(n-6)-4a(n-7)-a(n-8) for n>=11.

Edited by Dean Hickerson, Aug 21 2002

A074085 Coefficient of q^2 in nu(n), where nu(0) = 1, nu(1) = b and, for n >= 2, nu(n) = bnu(n-1) + lambda(1 + q + q^2 + ... + q^(n - 2))*nu(n-2) with (b,lambda) = (2,1).

Original entry on oeis.org

0, 0, 0, 0, 5, 24, 91, 308, 978, 2978, 8802, 25440, 72251, 202316, 559941, 1534548, 4170256, 11250630, 30158900, 80389600, 213204513, 562896832, 1480086111, 3877337556, 10123000126, 26347306474, 68378847990, 176994780672

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

nonn

The coefficient of q^0 is the Pell number A000129(n+1).

			The first 6 nu polynomials are nu(0) = 1, nu(1) = 2, nu(2) = 5, nu(3) = 12 + 2*q, nu(4) = 29 + 9*q + 5*q^2, nu(5) = 70 + 32q + 24*q^2 + 14*q^3 + 2*q^4, so the coefficients of q^2 are 0,0,0,0,5,24.

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Index entries for linear recurrences with constant coefficients, signature (6, -9, -4, 9, 6, 1).

Coefficients of q^0, q^1 and q^3 are in A000129, A074084 and A074086. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.

b=2; lambda=1; expon=2; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0,0}, LinearRecurrence[{6, -9, -4, 9, 6, 1}, {0, 0, 5, 24, 91, 308}, 30]] (* Jean-François Alcover, Dec 13 2018 *)

G.f.: (5*x^4 - 6*x^5 - 8*x^6 - 2*x^7)/(1 - 2*x - x^2)^3.

a(n) = 6*a(n-1) - 9*a(n-2) - 4*a(n-3) + 9*a(n-4) + 6*a(n-5) + a(n-6) for n >= 8.

Edited by Dean Hickerson, Aug 21 2002

A127532 Triangle read by rows: T(n,k) is the number of binary trees with n edges and jump-length equal to k (n >= 0, 0 <= k <= n-2).

Original entry on oeis.org

1, 2, 5, 12, 2, 29, 9, 4, 70, 32, 22, 8, 169, 102, 86, 56, 16, 408, 306, 296, 244, 144, 32, 985, 883, 949, 901, 712, 368, 64, 2378, 2480, 2908, 3056, 2822, 2096, 928, 128, 5741, 6828, 8633, 9830, 10074, 8976, 6144, 2304, 256, 13860, 18514, 25032, 30482, 33792

Offset: 0

Emeric Deutsch, Jan 18 2007

In the preorder traversal of a binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given binary tree is called the jump-length.
Rows 0 and 1 have one term each; row n (n >= 2) has n-1 terms.
Row sums are the Catalan numbers (A000108).
T(n,0) = A000129(n+1) (the Pell numbers).
T(n,1) = A074084(n).
Sum_{k>=0} k*T(n,k) = binomial(2n+1, n-3) + binomial(2n, n-3) = A127533(n).
The distribution of the statistic "number of jumps" is given in A127530.
The average jump distance in all binary trees with n edges is 2n(3n+5)(2n-1)/((n+3)(n+4)(3n+1)) (i.e., about 4 levels when n is large). The Krandick reference considers jump-length for full binary trees.

			Triangle starts:
   1;
   2;
   5;
  12,  2;
  29,  9,  4;
  70, 32, 22,  8;

W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.

Cf. A000108, A000129, A074084, A127530, A127533.

G:=(-1+2*t-2*t*z-sqrt(1-4*t*z+4*t^2*z^2-4*t*z^2))/2/(z^2-1+t-2*t*z+2*z): Gser:=simplify(series(G,z=0,17)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: 1;2;for n from 2 to 12 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form

G.f.: G = G(t,z) is given by (1 - t - 2z + 2tz - z^2)*G^2 - (1 - 2t + 2tz)*G - t = 0.

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A074082 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,1).

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A074086 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,1).

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A074083 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,1).

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A074085 Coefficient of q^2 in nu(n), where nu(0) = 1, nu(1) = b and, for n >= 2, nu(n) = b*nu(n-1) + lambda*(1 + q + q^2 + ... + q^(n - 2))*nu(n-2) with (b,lambda) = (2,1).

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A127532 Triangle read by rows: T(n,k) is the number of binary trees with n edges and jump-length equal to k (n >= 0, 0 <= k <= n-2).

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A074082 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,1).

A074086 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,1).

A074083 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,1).

A074085 Coefficient of q^2 in nu(n), where nu(0) = 1, nu(1) = b and, for n >= 2, nu(n) = bnu(n-1) + lambda(1 + q + q^2 + ... + q^(n - 2))*nu(n-2) with (b,lambda) = (2,1).