A052952 -id:A052952 - cp's OEIS frontend

A054450 Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials).

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 4, 4, 1, 1, 8, 8, 5, 5, 1, 1, 13, 12, 12, 6, 6, 1, 1, 21, 21, 17, 17, 7, 7, 1, 1, 34, 33, 33, 23, 23, 8, 8, 1, 1, 55, 55, 50, 50, 30, 30, 9, 9, 1, 1, 89, 88, 88, 73, 73, 38, 38, 10, 10, 1, 1, 144, 144, 138, 138, 103, 103, 47, 47, 11, 11, 1, 1

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is Fib(z)/(1-x*z/(1-z^2)) where Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).

This is the first member of the family of Riordan-type matrices obtained from the unsigned convolution matrix A049310 by repeated application of the partial row sums procedure.

			Triangle begins as:
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1,  1;
   5,  4,  4,  1,  1;
   8,  8,  5,  5,  1,  1;
  13, 12, 12,  6,  6,  1,  1;
  21, 21, 17, 17,  7,  7,  1,  1;
  34, 33, 33, 23, 23,  8,  8,  1,  1;
  55, 55, 50, 50, 30, 30,  9,  9,  1, 1;
  89, 88, 88, 73, 73, 38, 38, 10, 10, 1, 1;
  ...
Fourth row polynomial (n=3): p(3,x) = 3 + 3*x + x^2 + x^3.

Cf. A000027, A000045, A029907 (row sums), A038793, A038794, A038795, A038796.

Cf. A049310, A052952, A053121, A054451, A054453, A099571, A099572, A152948.

A049310:= func< n,k | ((n+k) mod 2) eq 0 select (-1)^(Floor((n+k)/2)+k)*Binomial(Floor((n+k)/2), k) else 0 >;
A054450:= func< n,k | (&+[Abs(A049310(n,j)): j in [k..n]]) >;
[A054450(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 25 2022

A049310[n_, k_]:= A049310[n, k]= If[n<0, 0, If[k==n, 1, A049310[n-1, k-1] - A049310[n-2, k] ]];
A054450[n_, k_]:= A054450[n, k]= Sum[Abs[A049310[n,j]], {j,k,n}];
Table[A054450[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 25 2022 *)

@CachedFunction
def A049310(n, k):
    if (n<0): return 0
    elif (k==n): return 1
    else: return A049310(n-1, k-1) - A049310(n-2, k)
def A054450(n,k): return sum( abs(A049310(n,j)) for j in (k..n) )
flatten([[A054450(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 25 2022

T(n, m) = Sum_{k=m..n} |A049310(n, k)| (sequence of partial row sums in column m).

Column m recursion: T(n, m) = Sum_{j=m..n} T(j-1, m)*|A049310(n-j, 0)| + |A049310(n, m)|, n >= m >= 0, a(n, m) := 0 if n

T(n, 0) = A000045(n+1).

T(n, 1) = A052952(n-1).

T(n, 2) = A054451(n-2).

Sum_{k=0..n} T(n, k) = A029907(n) = A054453(n, 0).

G.f. for column m: Fib(x)*(x/(1-x^2))^m, m >= 0, with Fib(x) = g.f. A000045(n+1).

The corresponding square array has T(n, k) = Sum_{j=0..floor(k/2)} binomial(n+k-j, j). - Paul Barry, Oct 23 2004

From G. C. Greubel, Jul 25 2022: (Start)

T(n, 3) = A099571(n-3).

T(n, 4) = A099572(n-4).

T(n, n) = T(n, n-1) = A000012(n).

T(n, n-2) = A000027(n), n >= 2.

T(n, n-3) = A000027(n), n >= 3.

T(n, n-4) = A152948(n), n >= 4.

T(n, n-5) = A152948(n), n >= 5.

T(n, n-6) = A038793(n), n >= 6.

T(n, n-8) = A038794(n), n >= 8.

T(n, n-10) = A038795(n), n >= 10.

T(n, n-12) = A038796(n), n >= 12. (End)

A054451 Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).

Original entry on oeis.org

1, 1, 4, 5, 12, 17, 33, 50, 88, 138, 232, 370, 609, 979, 1596, 2575, 4180, 6755, 10945, 17700, 28656, 46356, 75024, 121380, 196417, 317797, 514228, 832025, 1346268, 2178293, 3524577, 5702870, 9227464, 14930334, 24157816, 39088150, 63245985, 102334135

Offset: 0

Wolfdieter Lang, Apr 27 2000

Equals triangle A173284 * [1, 2, 3, ...]. - Gary W. Adamson, Mar 03 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-3,1,1).

Cf. A054450, A049310, A000045, A052952.
Cf. A007382.
Cf. A173284.

BB:=1/(1-k^2)^2/(1-k-k^2): seq(coeff(series(BB, k, n+1), k, n), n=0..50); # Zerinvary Lajos, May 16 2007

LinearRecurrence[{1,3,-2,-3,1,1},{1,1,4,5,12,17},40] (* Harvey P. Dale, Oct 06 2024 *)

Vec(-1/((x-1)^2*(x+1)^2*(x^2+x-1)) + O(x^100)) \\ Colin Barker, Jun 14 2015

a(n) = A054450(n+2, 2).
G.f.: Fib(x)/(1-x^2)^2, with Fib(x)=1/(1-x-x^2) = g.f. A000045 (Fibonacci numbers without 0).
a(2*k) = A027941(k)= F(2*k+3)-1; a(2*k+1)= F(2*(k+2))-(k+2)= A054452(k), k >= 0.
a(n-2) = Fibonacci(n+1) - binomial(n-floor(n/2), floor(n/2)), or a(n-2) = Sum_{i=0..floor(n/2)-1} binomial(n-i, i). - Jon Perry, Mar 18 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, k). - Paul Barry, Oct 23 2004

More terms from James Sellers, Apr 28 2000

A054452 Partial sums of A027941(n-1) with a(-1) = 0.

Original entry on oeis.org

0, 0, 1, 5, 17, 50, 138, 370, 979, 2575, 6755, 17700, 46356, 121380, 317797, 832025, 2178293, 5702870, 14930334, 39088150, 102334135, 267914275, 701408711, 1836311880, 4807526952, 12586269000, 32951280073, 86267571245, 225851433689, 591286729850

Offset: 0

Wolfdieter Lang, Apr 27 2000

Colin Barker, Table of n, a(n) for n = 0..1000
László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (5th line of Table 1 is a(n-2)).
A. Shriki and O. Liba, Polygons with Fibonacci Number Coordinates: Problem B-1167, Fib. Quart. 54,2 May 2016, p. 180-181.
Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-1).

Cf. A027941, A054451, A001906, A052952.

I:=[0,0,1,5]; [n le 4 select I[n] else 5*Self(n-1)-8*Self(n-2)+5*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Mar 26 2015

a[0]:=0: a[1]:=1: for n from 2 to 50 do a[n] := 3*a[n-1]-a[n-2] od: seq(a[n]-n, n=0..27); # Zerinvary Lajos, Mar 20 2008
with(combinat): seq(fibonacci(2*n)-n, n=0..27); # Zerinvary Lajos, Jun 19 2008
g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)-n), n=0..27); # Zerinvary Lajos, Mar 22 2009

CoefficientList[Series[x^2 / ((1 - x)^2 (1 - 3 x + x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 26 2015 *)

makelist(sum(fib(k)*binomial(n+1,k+2),k,0,n),n,0,20); /* Vladimir Kruchinin, Oct 21 2016 */

concat(vector(2), Vec(x^2/((1-x)^2*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Jan 28 2017

[(lucas_number1(n, 3, 1)-lucas_number1(n, 2, 1)) for n in range(1, 28)]# Zerinvary Lajos, Mar 13 2009

a(n) = +5*a(n-1) -8*a(n-2) +5*a(n-3) -1*a(n-4).
G.f.: x^2/((1-x)^2*(1-3*x+x^2)).
a(n) = Sum_{k=0..n} A027941(k-1) = F(2*n)-n = A054450(2*n-1, 2) = A054451(2*n-3).
G.f.: x^2*Fibe(x)/(1-x)^2, with Fibe(x) := 1/(1-3*x+x^2) = g.f. A001906(n+1) (Fibonacci numbers F(2(n+1))).
Fourth diagonal of array defined by T(i, 1) = T(1, j) = 1, T(i, j) = Max(T(i-1, j) + T(i-1, j-1); T(i-1, j-1) + T(i, j-1)). - Benoit Cloitre, Aug 05 2003
a(n) = Sum_{k=0..n-2} binomial(2*n-k-1, k). - Johannes W. Meijer, Aug 12 2013
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} binomial(i+j, i-j). - Wesley Ivan Hurt, Mar 25 2015
a(n) = Sum_{k=0..n} (binomial(n+1,k+2)*Fibonacci(k)). - Vladimir Kruchinin, Oct 21 2016
a(n) = (-((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n)/sqrt(5) - n. - Colin Barker, Jan 28 2017

More terms from James Sellers, Apr 28 2000

a(0) added by Arkadiusz Wesolowski, Jun 07 2011

A112552 A modified Chebyshev transform of the second kind.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, 0, -3, 0, 1, 3, 0, -4, 0, 1, 0, 6, 0, -5, 0, 1, -4, 0, 10, 0, -6, 0, 1, 0, -10, 0, 15, 0, -7, 0, 1, 5, 0, -20, 0, 21, 0, -8, 0, 1, 0, 15, 0, -35, 0, 28, 0, -9, 0, 1, -6, 0, 35, 0, -56, 0, 36, 0, -10, 0, 1, 0, -21, 0, 70, 0, -84, 0, 45, 0, -11, 0, 1, 7, 0, -56, 0, 126, 0, -120, 0, 55, 0, -12, 0, 1

Offset: 0

Paul Barry, Sep 13 2005

Row sums are A112553.
Inverse is A112554.
Riordan array product (1/(1+x^2), x)*(1/(1+x^2), x/(1+x^2)).

			Triangle begins as:
   1;
   0,   1;
  -2,   0,   1;
   0,  -3,   0,   1;
   3,   0,  -4,   0,   1;
   0,   6,   0,  -5,   0,   1;
  -4,   0,  10,   0,  -6,   0,  1;
   0, -10,   0,  15,   0,  -7,  0,  1;
   5,   0, -20,   0,  21,   0, -8,  0,   1;
   0,  15,   0, -35,   0,  28,  0, -9,   0,   1;
  -6,   0,  35,   0, -56,   0, 36,  0, -10,   0, 1;
   0, -21,   0,  70,   0, -84,  0, 45,   0, -11, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A049310, A052952, A112553, A112554, A128174.

[(-1)^Floor((n-k)/2)*((1+(-1)^(n+k))/2)*Binomial(Floor((n+k+2)/2), k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 13 2022

Table[(-1)^Floor[(n-k)/2]*((1+(-1)^(n+k))/2)*Binomial[(n+k+2)/2, k+1], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 13 2022 *)

flatten([[(-1)^floor((n-k)/2)*((1+(-1)^(n+k))/2)*binomial((n+k+2)/2, k+1) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jan 13 2022

Riordan array (1/(1+x^2)^2, x/(1+x^2)).
T(n, k) = (-1)^floor((n-k)/2)*Sum_{j=0..n} (1+(-1)^(n-j))*(1+(-1)^(j-k))*binomial((j+k)/2, k)/4.
Unsigned triangle = A128174 * A149310, as infinite lower triangular matrices, with row sums A052952: (1, 1, 3, 4, 8, 12, 21, 33, ...). - Gary W. Adamson, Oct 28 2007
T(n, k) = (-1)^floor((n-k)/2)*((1 + (-1)^(n+k))/2)*binomial((n+k+2)/2, k+1). - G. C. Greubel, Jan 13 2022
T(n,k) = A049310(n+1,k+1) . - R. J. Mathar, Feb 07 2024

A303682 T(n,k) = Number of n X k 0..1 arrays with every element unequal to 0, 1 or 3 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 5, 7, 7, 5, 8, 17, 9, 17, 8, 13, 31, 15, 15, 31, 13, 21, 49, 25, 28, 25, 49, 21, 34, 103, 39, 44, 44, 39, 103, 34, 55, 193, 57, 64, 76, 64, 57, 193, 55, 89, 327, 87, 90, 110, 110, 90, 87, 327, 89, 144, 641, 137, 132, 150, 177, 150, 132, 137, 641, 144, 233, 1207

Offset: 1

R. H. Hardin, Apr 28 2018

Table starts
..1...2...3...5...8..13..21...34...55...89..144..233..377..610...987..1597
..2...5...7..17..31..49.103..193..327..641.1207.2129.4039.7585.13687.25585
..3...7...9..15..25..39..57...87..137..215..329..503..777.1207..1865..2871
..5..17..15..28..44..64..90..132..204..312..466..688.1024.1540..2318..3472
..8..31..25..44..76.110.150..212..306..458..684.1006.1462.2132..3122..4586
.13..49..39..64.110.177.245..335..467..657..949.1391.2035.2937..4221..6079
.21.103..57..90.150.245.367..509..697..957.1311.1851.2671.3869..5547..7907
.34.193..87.132.212.335.509..774.1086.1478.1984.2674.3730.5314..7588.10760
.55.327.137.204.306.467.697.1086.1684.2354.3144.4174.5578.7696.10808.15232

			Some solutions for n=5, k=4
..0..0..0..0. .0..0..0..0. .0..1..1..0. .0..1..1..1. .0..0..1..1
..0..0..0..0. .0..0..0..0. .1..1..1..1. .1..1..1..1. .0..1..1..1
..0..0..0..0. .0..0..0..0. .1..1..1..1. .1..1..1..1. .1..1..1..1
..1..0..0..0. .0..0..0..0. .1..1..1..1. .1..1..1..1. .1..1..1..1
..1..1..0..0. .0..0..0..0. .1..1..1..1. .0..1..1..0. .0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..1624

Column 1 is A000045(n+1).

Column 3 is A052952(n+4) for n>1.

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +4*a(n-3) -2*a(n-4) for n>5
k=3: a(n) = a(n-1) +2*a(n-4) for n>5
k=4: a(n) = a(n-1) +a(n-4) +a(n-5) for n>8
k=5: a(n) = a(n-1) +a(n-4) +a(n-6) for n>10
k=6: a(n) = a(n-1) +a(n-4) +a(n-7) for n>12
k=7: a(n) = a(n-1) +a(n-4) +a(n-8) for n>14

A099572 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+4, k).

Original entry on oeis.org

1, 1, 6, 7, 23, 30, 73, 103, 211, 314, 581, 895, 1560, 2455, 4135, 6590, 10890, 17480, 28590, 46070, 74946, 121016, 196326, 317342, 514123, 831465, 1346148, 2177613, 3524441, 5702054, 9227311, 14929365, 24157645, 39087010, 63245795, 102332805

Offset: 0

Paul Barry, Oct 23 2004

Fifth column of triangle A054450. In general Sum_{k=0..floor(n/2)} binomial(n-k+r, k), r>=0, will have g.f. 1/((1-x^2)^r*(1-x-x^2)) and for r>0, a(n) = Sum_{k=0..n} Fibonacci(n-k+1)*binomial(k/2+r-1, r-1)*(1+(-1)^k)/2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-10,6,10,-4,-5,1,1).

Cf. A000045, A054450, A054451, A052952, A099571.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x^2)^4*(1-x-x^2)) )); // G. C. Greubel, Jul 25 2022

Table[Fibonacci(n+5) +(-1)^n*(n^3+9*n^2+35*n+33)/96 -(n^3+21*n^2+155*n+417)/96, {n,0,40}] (* G. C. Greubel, Jul 25 2022 *)

[fibonacci(n+5) + (-1)^n*(n^3+9*n^2+35*n+33)/96 - (n^3+21*n^2+155*n + 417)/96 for n in (0..40)] # G. C. Greubel, Jul 25 2022

G.f.: 1/((1-x^2)^4*(1-x-x^2)). - corrected by R. J. Mathar, Feb 20 2011
a(n) = Sum_{k=0..n} Fibonacci(n-k+1)*binomial(k/2+3, 3)*((1+(-1)^k)/2).
a(n) = Fibonacci(n+5) + (-1)^n*(n^3 + 9*n^2 + 35*n + 33)/96 - (n^3 + 21*n^2 + 155*n + 417)/96. - G. C. Greubel, Jul 25 2022

A131326 Row sums of A131325.

Original entry on oeis.org

1, 3, 4, 9, 13, 24, 37, 63, 100, 165, 265, 432, 697, 1131, 1828, 2961, 4789, 7752, 12541, 20295, 32836, 53133, 85969, 139104, 225073, 364179, 589252, 953433, 1542685, 2496120, 4038805, 6534927, 10573732, 17108661, 27682393, 44791056, 72473449, 117264507

Offset: 0

Gary W. Adamson, Jun 28 2007

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A131325, A049310, A062114, A052952, A022086.

Vec((1+2*x-x^2) / ((x-1)*(1+x)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Jul 12 2017

a(n) = A062114(n+1)+A052952(n-1).
G.f. ( 1+2*x-x^2 ) / ( (x-1)*(1+x)*(x^2+x-1) ).  - R. J. Mathar, Aug 12 2012
a(n) = 3*Fibonacci(n+1)-2 if n even, a(n) = 3*Fibonacci(n+1) if n odd. - R. J. Mathar, Aug 12 2012
From Colin Barker, Jul 12 2017: (Start)
a(n) = (3*2^(-n-1)*((1 + sqrt(5))^(n+1) - (1 - sqrt(5))^(n+1))) / sqrt(5) - 2 for n even.
a(n) = (3*2^(-n-1)*((1 + sqrt(5))^(n+1) - (1 - sqrt(5))^(n+1))) / sqrt(5) for n odd.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.
(End)

A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 2, 1, 8, 3, 1, 13, 5, 2, 21, 8, 3, 1, 34, 13, 5, 2, 1, 55, 21, 8, 3, 1, 89, 34, 13, 5, 2, 1, 144, 55, 21, 8, 3, 1, 233, 89, 34, 13, 5, 2, 1, 377, 144, 55, 21, 8, 3, 1, 610, 233, 89, 34, 13, 5, 2, 1

Offset: 0

Gary W. Adamson, Feb 14 2010

The row sums equal A052952.
Let the triangle = M. Then lim_{n->infinity} M^n = A173285 as a left-shifted vector.
A173284 * [1, 2, 3, ...] = A054451: (1, 1, 4, 5, 12, 17, 33, ...). - Gary W. Adamson, Mar 03 2010
From Johannes W. Meijer, Sep 05 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A104762.
The diagonal sums lead to A004695. (End)

			First few rows of the triangle:
    1;
    1;
    2,   1;
    3,   1;
    5,   2,  1;
    8,   3,  1;
   13,   5,  2,  1;
   21,   8,  3,  1;
   34,  13,  5,  2,  1;
   55,  21,  8,  3,  1;
   89,  34, 13,  5,  2, 1;
  144,  55, 21,  8,  3, 1;
  233,  89, 34, 13,  5, 2, 1;
  377, 144, 55, 21,  8, 3, 1;
  610, 233, 89, 34, 13, 5, 2, 1;
  ...

Cf. A000045, A004695, A052952, A054451, A173285.

Cf. (Similar triangles) A008315 (Catalan), A011973 (Pascal), A102541 (Losanitsch), A122196 (Fractal), A122197 (Fractal), A128099 (Pell-Jacobsthal), A152198, A152204, A207538, A209634.

T := proc(n, k): if n<0 then return(0) elif k < 0 or k > floor(n/2) then return(0) else combinat[fibonacci](n-2*k+1) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..14); # Johannes W. Meijer, Sep 05 2013

Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.
From Johannes W. Meijer, Sep 05 2013: (Start)
T(n,k) = A000045(n-2*k+1), n >= 0 and 0 <= k <= floor(n/2).
T(n,k) = A104762(n-k, k). (End)

Term a(15) corrected by Johannes W. Meijer, Sep 05 2013

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A062114 a(n) = 2*Fibonacci(n) - (1 - (-1)^n)/2.

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A007179 Dual pairs of integrals arising from reflection coefficients.

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A054450 Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials).

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A054451 Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).

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Maple

Mathematica

PARI

Formula

Extensions

A054452 Partial sums of A027941(n-1) with a(-1) = 0.

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Magma

Maple

Mathematica

Maxima

PARI

Sage

Formula

Extensions

A112552 A modified Chebyshev transform of the second kind.

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