A014983 -id:A014983 - cp's OEIS frontend

A074088 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)=(2,3).

0, 0, 0, 0, 21, 120, 585, 2508, 10122, 39042, 145974, 532704, 1907451, 6725004, 23407287, 80591148, 274899288, 930128646, 3124838844, 10432356000, 34634029713, 114403303008, 376184538165, 1231890463020, 4018920819606

Offset: 0

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

The coefficient of q^0 is A014983(n+1).

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=7, nu(3)=20+6q, nu(4)=61+33q+21q^2, nu(5)=182+144q+120q^2+78q^3+18q^4, so the coefficients of q^2 are 0,0,0,0,21,120.

G. C. Greubel, Table of n, a(n) for n = 0..1000
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,-3,-28,9,54,27).

Coefficients of q^0, q^1 and q^3 are in A014983, A074087 and A074089. Related sequences with other values of b and lambda are in A074082-A074086.

I:=[0,0,21,120,585,2508]; [0,0] cat [n le 6 select I[n] else 6*Self(n-1) -3*Self(n-2) -28*Self(n-3) +9*Self(n-4) +54*Self(n-5) +27*Self(n-6): n in [1..30]]; // G. C. Greubel, May 26 2018

b=2; lambda=3; 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,-3,-28,9,54,27},{0,0,21,120,585,2508},40]] (* Harvey P. Dale, Apr 28 2012 *)

x='x+O('x^30); concat([0,0,0,0], Vec((21*x^4 -6*x^5 -72*x^6 -54*x^7)/(1-2*x-3*x^2)^3)) \\ G. C. Greubel, May 26 2018

G.f.: (21*x^4 -6*x^5 -72*x^6 -54*x^7)/(1-2*x-3*x^2)^3.

a(n) = 6*a(n-1) -3*a(n-2) -28*a(n-3) +9*a(n-4) +54*a(n-5) +27*a(n-6) for n>=8.

Edited by Dean Hickerson, Aug 21 2002

A248811 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+3)^k for 0 <= k <= n.

Original entry on oeis.org

1, -2, 1, 7, -5, 1, -20, 22, -8, 1, 61, -86, 46, -11, 1, -182, 319, -224, 79, -14, 1, 547, -1139, 991, -461, 121, -17, 1, -1640, 3964, -4112, 2374, -824, 172, -20, 1, 4921, -13532, 16300, -11234, 4846, -1340, 232, -23, 1, -14762, 45517, -62432, 50002, -25772, 8866, -2036, 301, -26, 1, 44287, -151313, 232813, -212438, 127318, -52370, 14974, -2939, 379, -29, 1

Offset: 0

Derek Orr, Oct 14 2014

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+3)^0 + A_1*(x+3)^1 + A_2*(x+3)^2 + ... + A_n*(x+3)^n. This sequence gives A_0, ..., A_n as the entries in the n-th row of this triangle, starting at n = 0.

			       1;
      -2,       1;
       7,      -5,      1;
     -20,      22,     -8,       1;
      61,     -86,     46,     -11,      1;
    -182,     319,   -224,      79,    -14,      1;
     547,   -1139,    991,    -461,    121,    -17,     1;
   -1640,    3964,  -4112,    2374,   -824,    172,   -20,     1;
    4921,  -13532,  16300,  -11234,   4846,  -1340,   232,   -23,   1;
  -14762,   45517, -62432,   50002, -25772,   8866, -2036,   301, -26,   1;
   44287, -151313, 232813, -212438, 127318, -52370, 14974, -2939, 379, -29, 1;

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

Cf. A193843, A077925, A191008, A014983.

[[(&+[(-3)^(j-k)*Binomial(j,k): j in [0..n]]): k in [0..n]]: n in [0..20]]; // G. C. Greubel, May 27 2018

T[n_, k_]:= Sum[(-3)^(j-k)*Binomial[j,k], {j,0,n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 27 2018 *)

for(n=0,20,for(k=0,n,print1(sum(i=0,n,((-3)^(i-k)* binomial(i, k)) ),", ")))

T(n,n-1) = -3*n + 1 for n > 0.
T(n,0) = A014983(n+1).
T(n,1) = (-1)^(n+1)*A191008(n-1).
Row n sums to A077925(n).

A345035 a(n) = Sum_{k=1..n} (-3)^(floor(n/k) - 1).

Original entry on oeis.org

1, -2, 11, -28, 81, -234, 739, -2216, 6545, -19594, 59139, -177408, 531181, -1593614, 4783799, -14351032, 43044597, -129133854, 387426799, -1162281332, 3486765521, -10460293354, 31381119459, -94143358440, 282429356977, -847288080362, 2541866366171

Offset: 1

Seiichi Manyama, Jun 06 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..2000

Column k=3 of A345033.

Cf. A014983, A345029.

a[n_] := Sum[(-3)^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 30] (* Amiram Eldar, Jun 06 2021 *)

PARI
```
a(n) = sum(k=1, n, (-3)^(n\k-1));
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k*(1-x^k)/(1+3*x^k))/(1-x))

G.f.: (1/(1 - x)) * Sum_{k>=1} x^k * (1 - x^k)/(1 + 3*x^k).

A124137 A signed aerated and skewed version of A038137.

Original entry on oeis.org

1, 0, 1, -1, 0, 2, 0, -2, 0, 3, 1, 0, -5, 0, 5, 0, 3, 0, -10, 0, 8, -1, 0, 9, 0, -20, 0, 13, 0, -4, 0, 22, 0, -38, 0, 21, 1, 0, -14, 0, 51, 0, -71, 0, 34, 0, 5, 0, -40, 0, 111, 0, -130, 0, 55, -1, 0, 20, 0, -105, 0, 233, 0, -235, 0, 89

Offset: 0

Philippe Deléham, Nov 30 2006

			Triangle begins:
1;
0, 1;
-1, 0, 2;
0, -2, 0, 3;
1, 0, -5, 0, 5;
0, 3, 0, -10, 0, 8;
-1, 0, 9, 0, -20, 0, 13;
0, -4, 0, 22, 0, -38, 0, 21;
1, 0, -14, 0, 51, 0, -71, 0, 34;
0, 5, 0, -40, 0, 111, 0, -130, 0, 55;

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

Cf. A038137, A208336, A000045, A001629, A001628, A001872, A001873

T[0, 0]:= 1; T[n_, n_]:= Fibonacci[n + 1]; T[n_, k_]:= T[n, k] = If[k < 0 || n < k, 0, T[n - 1, k - 1] + T[n - 2, k - 2] - T[n - 2, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten  (* G. C. Greubel, May 27 2018 *)

{T(n,k) = if(n==0 && k==0, 1, if(k==n, fibonacci(n+1), if(k<0 || nG. C. Greubel, May 27 2018

T(n,k) = T(n-1,k-1) + T(n-2,k-2) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if nA000045(n+1).

Sum_{0<=k<=n} x^k*T(n,k)= A014983(n+1), A033999(n), A056594(n), A000012(n), A015518(n+1), A015525(n+1) for x=-2, -1, 0, 1, 2, 3 respectively.

Corrected and extended by Philippe Deléham, Apr 05 2012

A072985 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n >= 2, nu(n) = bnu(n-1) + lambda(n-1)_q*nu(n-2) with (b,lambda)=(2,3), where (n)_q = (1+q+...+q^(n-1)) and q is a root of unity.

Original entry on oeis.org

1, 2, 7, 6, 21, 18, 63, 54, 189, 162, 567, 486, 1701, 1458, 5103, 4374, 15309, 13122, 45927, 39366, 137781, 118098, 413343, 354294, 1240029, 1062882, 3720087, 3188646, 11160261, 9565938, 33480783, 28697814, 100442349, 86093442

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Instead of listing the coefficients of the highest power of q in each nu(n), if we list the coefficients of the smallest power of q (i.e., constant terms), we get a sequence of weighted Fibonacci numbers described by f(0)=1, f(1)=1, for n >= 2, f(n) = 2*f(n-1) + 3*f(n-2).

			nu(0) = 1;
nu(1) = 2;
nu(2) = 7;
nu(3) = 20 + 6q;
nu(4) = 61 + 33q + 21q^2;
nu(5) = 182 + 144q + 120q^2 + 78q^3 + 18q^4;
nu(6) = 547 + 570q + 585q^2 + 501q^3 + 381q^4 + 162q^5 + 63q^6; ...
The coefficients of the highest power of q give this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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 (0,3).

Cf. A014983.

[1] cat [(1/6)*(13+(-1)^n)*3^Floor(n/2): n in [1..40]]; // Vincenzo Librandi, Jul 20 2013

CoefficientList[Series[-(1 + 2 x + 4 x^2) / (-1 + 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)
Join[{1}, LinearRecurrence[{0, 3}, {2, 7}, 33]] (* Jean-François Alcover, Sep 23 2017 *)

x='x+O('x^30); Vec((1+2*x+4*x^2)/(1-3*x^2)) \\ G. C. Greubel, May 26 2018

For given b and lambda, the recurrence relation is given by; t(0)=1, t(1)=b, t(2) = b^2+lambda and for n >= 3, t(n) = lambda*t(n-2).
G.f.: (1 + 2*x + 4*x^2)/(1-3*x^2). - R. J. Mathar, Dec 05 2007
a(n) = 3*a(n-2) for n>2. - Ralf Stephan, Jul 19 2013
a(n) = (1/6)*(13 + (-1)^n)*3^floor(n/2) for n>0. - Ralf Stephan, Jul 19 2013

More terms from R. J. Mathar, Dec 05 2007

A268413 a(n) = Sum_{k = 0..n} (-1)^k*14^k.

Original entry on oeis.org

1, -13, 183, -2561, 35855, -501969, 7027567, -98385937, 1377403119, -19283643665, 269971011311, -3779594158353, 52914318216943, -740800455037201, 10371206370520815, -145196889187291409, 2032756448622079727, -28458590280709116177, 398420263929927626479

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

Alternating sum of powers of 14.

More generally, the ordinary generating function for the Sum_{k = 0..n} (-1)^k*m^k is 1/(1 + (m - 1)*x - m*x^2). Also, Sum_{k = 0..n} (-1)^k*m^k = ((-1)^n*m^(n + 1) + 1)/(m + 1).

G. C. Greubel, Table of n, a(n) for n = 0..870
Index entries for linear recurrences with constant coefficients, signature (-13,14).

Cf. A001023, A033999.

Cf. similar sequences of the type Sum_{k=0..n} (-1)^k*m^k: A059841 (m=1), A077925 (m=2), A014983 (m=3), A014985 (m=4), A014986 (m=5), A014987 (m=6), A014989 (m=7), A014990 (m=8), A014991 (m=9), A014992 (m=10), A014993 (m=11), A014994 (m=12), A015000 (m=13), this sequence (m=14), A239284 (m=15).

I:=[1,-19]; [n le 2 select I[n] else -13*Self(n-1) +14*Self(n-2): n in [1..30]]; // G. C. Greubel, May 26 2018

Table[((-1)^n 14^(n + 1) + 1)/15, {n, 0, 18}]
LinearRecurrence[{-13, 14}, {1, -13}, 19]
Table[Sum[(-1)^k*14^k, {k, 0, n}], {n, 0, 18}]

x='x+O('x^30); Vec(1/(1 + 13*x - 14*x^2)) \\ G. C. Greubel, May 26 2018

G.f.: 1/(1 + 13*x - 14*x^2).
a(n) = ((-1)^n*14^(n + 1) + 1)/15.
a(n) = 1 - 14*a(n - 1) for n>0 and a(0)=1.
a(n) = Sum_{k = 0..n} A033999(k)*A001023(k).
Lim_{n -> infinity} a(n)/a(n + 1) = - 1/14.

A346083 Triangle, read by rows, defined by recurrence: T(n,k) = T(n-1,k-1) + (-1)^k * (2 * k + 1) * T(n-1,k) for 0 < k < n with initial values T(n,0) = T(n,n) = 1 for n >= 0 and T(i,j) = 0 if j < 0 or j > i.

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, 7, 3, 1, 1, -20, 22, -4, 1, 1, 61, 90, 50, 5, 1, 1, -182, 511, -260, 95, -6, 1, 1, 547, 2373, 2331, 595, 161, 7, 1, 1, -1640, 12412, -13944, 7686, -1176, 252, -8, 1, 1, 4921, 60420, 110020, 55230, 20622, 2100, 372, 9, 1, 1, -14762, 307021, -709720, 607090, -171612, 47922, -3480, 525, -10, 1

Offset: 0

Werner Schulte, Jul 04 2021

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :  0      1      2       3      4      5     6    7  8  9
=============================================================
  0 :  1
  1 :  1      1
  2 :  1     -2      1
  3 :  1      7      3       1
  4 :  1    -20     22      -4      1
  5 :  1     61     90      50      5      1
  6 :  1   -182    511    -260     95     -6     1
  7 :  1    547   2373    2331    595    161     7    1
  8 :  1  -1640  12412  -13944   7686  -1176   252   -8  1
  9 :  1   4921  60420  110020  55230  20622  2100  372  9  1
  etc.

Cf. A000012 (column 0 and main diagonal), A014983 (column 1), A181983 (1st subdiagonal), A002412 (2nd subdiagonal), A264851 (3rd subdiagonal without signs).

Cf. A051159.

from functools import cache
@cache
def T(n, k):
    if k == 0 or k == n: return 1
    return T(n-1, k-1) + (-1)**k*(2*k + 1)*T(n-1, k)
for n in range(10):
    print([T(n, k) for k in range(n+1)]) # Peter Luschny, Jul 22 2021

G.f. of column k >= 0: col(t,k) = Sum_{n >= k} T(n,k) * t^n = t^k / (Product_{i=0..k} (1 - (-1)^i * (2 * i + 1) * t)), i.e., col(t,k) = col(t,k-1) * t / (1 - (-1)^k * (2 * k + 1) * t) for k > 0.
Matrix inverse M = T^(-1) has row polynomials p(n,x) = Sum_{k=0..n} M(n,k) * x^k = Product_{i=1..n} (x + (-1)^i * (2 * i - 1)) for n >= 0 and empty product 1, i.e., p(n,x) = p(n-1,x) * (x + (-1)^n * (2 * n - 1)) for n > 0 with initial value p(0,x) = 1.
Conjecture: E.g.f. of column k >= 0: Sum_{n >= k} T(n,k) * t^n / (n!) = (Sum_{i=0..k} (-1)^(i * (i + 1 ) / 2) * binomial(k,floor((k - i) / 2)) * exp((-1)^i * (2 * i + 1) * t)) * (-1)^(k * (k - 1) / 2) / (4^k * (k!)), i.e., T(n,k) = (Sum_{i=0..k} (-1)^(i * (i + 1) / 2) * binomial(k,floor((k - i) / 2)) * ((-1)^i * (2 * i + 1))^n) * (-1)^(k * (k - 1) / 2) / (4^k * (k!)) for 0 <= k <= n.
Conjecture: E.g.f. of column k >= 0: Sum_{n >= k} T(n,k) * t^n / (n!) = exp(t) * (exp(4*t) - 1)^k / (4^k * (k!) * exp(4*t*floor((k+1)/2))), i.e., T(n,k) = (Sum_{i=0..k} (-1)^i * binomial(k,i) *(1 + 4*i - 4*floor((k+1)/2))^n) * (-1)^k / (4^k * (k!)) for 0 <= k <= n. Proved by Burkhard Hackmann and Werner Schulte (distinction of two cases: odd k, even k). - Werner Schulte, Aug 03 2021

cp's OEIS Frontend

A074088 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)=(2,3).

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A248811 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+3)^k for 0 <= k <= n.

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A345035 a(n) = Sum_{k=1..n} (-3)^(floor(n/k) - 1).

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A124137 A signed aerated and skewed version of A038137.

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A072985 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n >= 2, nu(n) = b*nu(n-1) + lambda*(n-1)_q*nu(n-2) with (b,lambda)=(2,3), where (n)_q = (1+q+...+q^(n-1)) and q is a root of unity.

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A268413 a(n) = Sum_{k = 0..n} (-1)^k*14^k.

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A346083 Triangle, read by rows, defined by recurrence: T(n,k) = T(n-1,k-1) + (-1)^k * (2 * k + 1) * T(n-1,k) for 0 < k < n with initial values T(n,0) = T(n,n) = 1 for n >= 0 and T(i,j) = 0 if j < 0 or j > i.

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A074088 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)=(2,3).

A072985 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n >= 2, nu(n) = bnu(n-1) + lambda(n-1)_q*nu(n-2) with (b,lambda)=(2,3), where (n)_q = (1+q+...+q^(n-1)) and q is a root of unity.