A112554 -id:A112554 - cp's OEIS frontend

A112552 A modified Chebyshev transform of the second kind.

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

A112553 Expansion of 1/( (1+x^2)*(1-x+x^2) ).

Original entry on oeis.org

1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0, 1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0, 1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0, 1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0, 1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0, 1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0

Offset: 0

Paul Barry, Sep 13 2005

Row sums of A112552.

			G.f. = 1 + x - x^2 - 2*x^3 + 2*x^5 + x^6 - x^7 - x^8 + x^12 + x^13 - x^14 - 2*x^15 + ... - _Michael Somos_, Feb 15 2024

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

Cf. A010892, A112552, A112554.

Join[{1}, PadRight[{}, 120, {1,-1,-2,0,2,1,-1,-1,0,0,0,1}]] (* G. C. Greubel, Jan 13 2022 *)
a[ n_] := {1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0}[[1 + Mod[n, 12]]]; (* Michael Somos, Feb 15 2024 *)

{a(n) = [1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0][1 + n%12]}; /* Michael Somos, Feb 15 2024 */

a(n) = (1/4)*Sum_{k=0..n} (-1)^floor((n-k)/2)*Sum_{j=0..n} (1+(-1)^(n-j))*(1+(-1)^(j-k)) *binomial((j+k)/2, k).
From G. C. Greubel, Jan 13 2022: (Start)
a(n) = Sum_{k=0..n} (-1)^floor((n-k)/2)*((1 + (-1)^(n+k))/2)*binomial((n+k+2)/2, k+1).
a(n + 12) = a(n). (End)
a(n) = -a(-4-n) for all n in Z. - Michael Somos, Feb 15 2024

A123514 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 321 exactly once (n>=3; 1<=k<=n-2).

Original entry on oeis.org

1, 0, 2, 4, 0, 3, 0, 10, 0, 4, 14, 0, 18, 0, 5, 0, 40, 0, 28, 0, 6, 48, 0, 81, 0, 40, 0, 7, 0, 150, 0, 140, 0, 54, 0, 8, 165, 0, 330, 0, 220, 0, 70, 0, 9, 0, 550, 0, 616, 0, 324, 0, 88, 0, 10, 572, 0, 1287, 0, 1040, 0, 455, 0, 108, 0, 11, 0, 2002, 0, 2548, 0, 1638, 0, 616, 0, 130, 0, 12

Offset: 3

Emeric Deutsch, Oct 13 2006

			T(4,2)=2 because we have 1432 and 3214 (also 4231 is an involution with 2 fixed points but contains twice the pattern 321: 421 and 431).
Triangle starts:
    1;
    0,   2;
    4,   0,    3;
    0,  10,    0,   4;
   14,   0,   18,   0,    5;
    0,  40,    0,  28,    0,   6;
   48,   0,   81,   0,   40,   0,   7;
    0, 150,    0, 140,    0,  54,   0,  8;
  165,   0,  330,   0,  220,   0,  70,  0,   9;
    0, 550,    0, 616,    0, 324,   0, 88,   0, 10;
  572,   0, 1287,   0, 1040,   0, 455,  0, 108,  0, 11;

G. C. Greubel, Rows n = 3..53 of the triangle, flattened
E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 493 and 498).

Cf. A003517, A112554, A123515, A191389.

A123514:= func< n,k | ((1+(-1)^(n-k))/(2*(n+1)))*k*(k+3)*Binomial(n+1, Floor((n-k-2)/2)) >;
[A123514(n,k): k in [1..n-2], n in [3..15]]; // G. C. Greubel, Jan 15 2022

T:=proc(n,k) if n-k mod 2 = 0 and k<=n then k*(k+3)*binomial(n+1,(n-k)/2-1)/(n+1) else 0 fi end: for n from 3 to 15 do seq(T(n,k),k=1..n-2) od; # yields sequence in triangular form

T[n_, k_]:= ((1+(-1)^(n-k))/2)*k*(k+3)*Binomial[n+1, (n-k-2)/2]/(n+1);
Table[T[n, k], {n, 3, 15}, {k, n-2}]//Flatten (* G. C. Greubel, Jan 15 2022 *)

def A123514(n,k): return ((1+(-1)^(n-k))/(2*(n+1)))*k*(k+3)*binomial(n+1, (n-k-2)//2)
flatten([[A123514(n,k) for k in (1..n-2)] for n in (3..15)]) # G. C. Greubel, Jan 15 2022

T(n,k) = k*(k+3)*binomial(n+1,(n-k-2)/2)/(n+1), for n>=3, 1<=k<=n-2, n-k even.
From G. C. Greubel, Jan 15 2022: (Start)
Sum_{k=1..n-2} T(n, k) = A191389(n+1).
Sum_{k=1..floor((n-1)/2)} T(n-k, k) = ((1-(-1)^n)/2)*(12/(n+9))*binomial(n+2, (n- 3)/2). (End)

A123515 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 231 exactly once (n>=4, 2<=k<=n-2).

Original entry on oeis.org

1, 0, 2, 2, 0, 3, 0, 8, 0, 4, 5, 0, 18, 0, 5, 0, 26, 0, 32, 0, 6, 12, 0, 75, 0, 50, 0, 7, 0, 76, 0, 164, 0, 72, 0, 8, 28, 0, 264, 0, 305, 0, 98, 0, 9, 0, 208, 0, 680, 0, 510, 0, 128, 0, 10, 64, 0, 840, 0, 1460, 0, 791, 0, 162, 0, 11, 0, 544, 0, 2480, 0, 2772, 0, 1160, 0, 200, 0, 12

Offset: 4

Emeric Deutsch, Oct 13 2006

Also the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 312 exactly once (n>=4, 2<=k<=n-2). Example: T(5,3)=2 because we have 15342 and 42315 (also the involution 52341 has 3 fixed points but it contains 3 times the pattern 312: 523, 524 and 534).

			T(5,3)=2 because we have 15342 and 42315 (also the involution 52341 has 3 fixed points but it contains 3 times the pattern 231: 231, 241 and 341).
Triangle starts:
   1;
   0,   2;
   2,   0,   3;
   0,   8,   0,    4;
   5,   0,  18,    0,    5;
   0,  26,   0,   32,    0,    6;
  12,   0,  75,    0,   50,    0,   7;
   0,  76,   0,  164,    0,   72,   0,    8;
  28,   0, 264,    0,  305,    0,  98,    0,   9;
   0, 208,   0,  680,    0,  510,   0,  128,   0,  10;
  64,   0, 840,    0, 1460,    0, 791,    0, 162,   0, 11;
   0, 544,   0, 2480,    0, 2772,   0, 1160,   0, 200,  0, 12;

G. C. Greubel, Rows n = 4..54 of the triangle, flattened
E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 492 and 498).

Cf. A045623, A120926, A123514, A112554.

T:=proc(n,k) if n>=4 and n+k mod 2 = 0 then (k-1)*2^((n-k-6)/2)*(binomial((n+k)/2-2,(n-k)/2-1)+2*binomial((n+k)/2-3, (n-k)/2-1)+binomial((n+k)/2-4,(n-k)/2-1)) else 0 fi end: for n from 4 to 16 do seq(T(n,k),k=2..n-2) od; # yields sequence in triangular form

T[n_, k_]:= ((1+(-1)^(n-k))/2)*2^((n-k-6)/2)*(k-1)* Sum[Binomial[2, j]*
  Binomial[(n+k-2*(j+2))/2, (n-k-2)/2], {j, 0, 2}];
Table[T[n, k], {n,4,16}, {k,2,n-2}]//Flatten (* G. C. Greubel, Jan 16 2022 *)

def A123515(n,k): return ((1+(-1)^(n+k))/2)*2^((n-k-6)/2)*(k-1)*sum( binomial(2, j)*binomial((n+k-2*j-2)/2, (n-k-2)/2) for j in (0..2) )
flatten([[A123515(n,k) for k in (2..n-2)] for n in (4..16)]) # G. C. Greubel, Jan 16 2022

T(n, k) = 2^((n-k-6)/2)*(k-1)*( binomial((n+k)/2-2, (n-k)/2-1) + 2*binomial((n+k)/2-3, (n-k)/2-1) + binomial((n+k)/2-4, (n-k)/2-1) ) for n>=4, n+k even; T(n,k) = 0 otherwise.
From G. C. Greubel, Jan 16 2022: (Start)
Sum_{k=2..n-4} T(n, k) = A045623(n).
Sum_{k=2..floor(n/2)} T(n-k+2, k) = (1/9)*[n=4] + (1+(-1)^n)*n*3^((n-8)/2). (End)

cp's OEIS Frontend

A112552 A modified Chebyshev transform of the second kind.

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A112553 Expansion of 1/( (1+x^2)*(1-x+x^2) ).

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A123514 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 321 exactly once (n>=3; 1<=k<=n-2).

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A123515 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 231 exactly once (n>=4, 2<=k<=n-2).

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Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

Sage

Formula