A008310 -id:A008310 - cp's OEIS frontend

A136389 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,3}(x) with 0 omitted (exponents in increasing order).

1, -3, 2, 3, -7, 4, -1, 9, -16, 8, -5, 25, -36, 16, 1, -19, 66, -80, 32, 7, -63, 168, -176, 64, -1, 33, -192, 416, -384, 128, -9, 129, -552, 1008, -832, 256, 1, -51, 450, -1520, 2400, -1792, 512, 11, -231, 1452, -4048, 5632, -3840, 1024, -1, 73, -912, 4424, -10496, 13056, -8192, 2048

Offset: 3

Milan Janjic, Mar 30 2008, revised Apr 05 2008

If U_n(x), T_n(x) are Chebyshev's polynomials then U_n(x)=P_{n,0}(x), T_n(x)=P_{n,1}(x).

Let n>=3 and k be of the same parity. Consider a set X consisting of (n+k)/2-3 blocks of the size 2 and an additional block of the size 3, then (-1)^((n-k)/2)a(n,k) is the number of n-3-subsets of X intersecting each block of the size 2.

			Rows are (1),(-3,2),(3,-7,4),(-1,9,-16,8),(-5,25,-36,16),...
since P_{3,3}=x^3, P_{4,3}=-3x^2+2x^4, P_{5,3}=3x-7x^3+4x^5,...

Michael De Vlieger, Table of n, a(n) for n = 3..10197 (rows 3 <= n <= 200, flattened).
Milan Janjic, Two enumerative functions.
Milan Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Cf. A008310, A053117.

if modp(n-k, 2)=0 then a[n, k]:=(-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-3, i)*binomial(n+k-3-2*i, n-3), i=0..(n+k)/2-3); end if;

DeleteCases[#, 0] &@ Flatten@ Table[(-1)^((n - k)/2)*Sum[(-1)^i*Binomial[(n + k)/2 - 3, i] Binomial[n + k - 3 - 2 i, n - 3], {i, 0, (n + k)/2 - 3}], {n, 3, 14}, {k, 0 + Boole[OddQ@ n], n, 2}] (* Michael De Vlieger, Jul 05 2019 *)

If n>=3 and k are of the same parity then a(n,k)= (-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-3, i)*binomial(n+k-3-2*i, n-3), i=0..(n+k)/2-3) and a(n,k)=0 if n and k are of different parity.

A136390 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,4}(x) with 0 omitted (exponents in increasing order).

Original entry on oeis.org

1, -4, 2, 6, -9, 4, -4, 16, -20, 8, 1, -14, 41, -44, 16, 6, -44, 102, -96, 32, -1, 26, -129, 248, -208, 64, -8, 96, -360, 592, -448, 128, 1, -42, 321, -968, 1392, -960, 256, 10, -180, 1002, -2528, 3232, -2048, 512, -1, 62, -681, 2972, -6448, 7424, -4352, 1024

Offset: 4

Milan Janjic, Mar 30 2008, revised Apr 05 2008

If U_n(x), T_n(x) are Chebyshev's polynomials then U_n(x)=P_{n,0}(x), T_n(x)=P_{n,1}(x).

Let n>=4 and k be of the same parity. Consider a set X consisting of (n+k)/2-4 blocks of the size 2 and an additional block of the size 4, then (-1)^((n-k)/2)a(n,k) is the number of n-4-subsets of X intersecting each block of the size 2.

			Rows are (1),(-4,2),(6,-9,4),(-4,16,-20,8),... since P_{4,4}=x^4, P_{5,4}=-4x^3+2x^5, P_{6,4}=6x^2-9x^4+4x^6,...

Michael De Vlieger, Table of n, a(n) for n = 4..10194 (rows 4 <= n <= 200, flattened).
Milan Janjic, Two enumerative functions.
M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Cf. A008310, A053117.

if modp(n-k, 2)=0 then a[n, k]:=(-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-4, i)*binomial(n+k-4-2*i, n-4), i=0..(n+k)/2-4); end if;

DeleteCases[#, 0] &@ Flatten@ Table[(-1)^((n - k)/2) * Sum[(-1)^i * Binomial[(n + k)/2 - 4, i] Binomial[n + k - 4 - 2 i, n - 4], {i, 0, (n + k)/2 - 4}], {n, 4, 14}, {k, 0 + Boole[OddQ@ n], n, 2}] (* Michael De Vlieger, Jul 05 2019 *)

If n>=4 and k are of the same parity then a(n,k)= (-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-4, i)*binomial(n+k-4-2*i, n-4), i=0..(n+k)/2-4) and a(n,k)=0 if n and k are of different parity.

A136397 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,5}(x) with 0 omitted (exponents in increasing order).

Original entry on oeis.org

1, -5, 2, 10, -11, 4, -10, 25, -24, 8, 5, -30, 61, -52, 16, -1, 20, -85, 146, -112, 32, -7, 70, -231, 344, -240, 64, 1, -34, 225, -608, 800, -512, 128, 9, -138, 681, -1560, 1840, -1088, 256, -1, 52, -501, 1970, -3920, 4192, -2304, 512

Offset: 5

Milan Janjic, Mar 30 2008, revised Apr 05 2008

If U_n(x), T_n(x) are Chebyshev's polynomials then U_n(x)=P_{n,0}(x), T_n(x)=P_{n,1}(x).

Let n>=5 and k be of the same parity. Consider a set X consisting of (n+k)/2-5 blocks of the size 2 and an additional block of the size 5, then (-1)^((n-k)/2)a(n,k) is the number of n-5-subsets of X intersecting each block of the size 2.

			Rows are (1),(-5,2),(10,-11,4),... since P_{5,5}=x^5, P_{6,5}=-5x^4+2x^6, P_{7,5}=10x^3-11x^5+4x^7,...

Michael De Vlieger, Table of n, a(n) for n = 5..10190 (rows 5 <= n <= 200, flattened).
Milan Janjic, Two enumerative functions.
M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Cf. A008310, A053117.

if modp(n-k, 2)=0 then a[n, k]:=(-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-5, i)*binomial(n+k-5-2*i, n-5), i=0..(n+k)/2-5); end if;

DeleteCases[#, 0] &@ Flatten@ Table[(-1)^((n - k)/2) * Sum[(-1)^i * Binomial[(n + k)/2 - 5, i] Binomial[n + k - 5 - 2 i, n - 5], {i, 0, (n + k)/2 - 5}], {n, 5, 14}, {k, 0 + Boole[OddQ@ n], n, 2}] (* Michael De Vlieger, Jul 05 2019 *)

If n>=5 and k are of the same parity then a(n,k)= (-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-5, i)*binomial(n+k-5-2*i, n-5), i=0..(n+k)/2-5) and a(n,k)=0 if n and k are of different parity.

A136398 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,6}(x) with 0 omitted (exponents in increasing order).

Original entry on oeis.org

1, -6, 2, 15, -13, 4, -20, 36, -28, 8, 15, -55, 85, -60, 16, -6, 50, -146, 198, -128, 32, 1, -27, 155, -377, 456, -272, 64, 8, -104, 456, -952, 1040, -576, 128, -1, 43, -363, 1289, -2360, 2352, -1216, 256, -10, 190, -1182, 3530, -5760, 5280, -2560, 512

Offset: 6

Milan Janjic, Mar 30 2008, revised Apr 05 2008

If U_n(x), T_n(x) are Chebyshev's polynomials then U_n(x)=P_{n,0}(x), T_n(x)=P_{n,1}(x).

Let n>=6 and k be of the same parity. Consider a set X consisting of (n+k)/2-6 blocks of the size 2 and an additional block of the size 6, then (-1)^((n-k)/2)a(n,k) is the number of n-6-subsets of X intersecting each block of the size 2.

			Rows are (1),(-6,2),(15,-13,4),... since P_{6,6}=x^6, P_{7,6}=-6x^5+2x^7, P_{8,6}=15x^4-13x^6+4x^8,...

Michael De Vlieger, Table of n, a(n) for n = 6..10185 (rows 6 <= n <= 200, flattened).
Milan Janjic, Two enumerative functions.
M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Cf. A008310, A053117.

if modp(n-k, 2)=0 then a[n, k]:=(-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-6, i)*binomial(n+k-6-2*i, n-6), i=0..(n+k)/2-6); end if;

DeleteCases[#, 0] &@ Flatten@ Table[(-1)^((n - k)/2) * Sum[(-1)^i * Binomial[(n + k)/2 - 6, i] Binomial[n + k - 6 - 2 i, n - 6], {i, 0, (n + k)/2 - 6}], {n, 6, 15}, {k, 0 + Boole[OddQ@ n], n, 2}] (* Michael De Vlieger, Jul 05 2019 *)

If n>=6 and k are of the same parity then a(n,k)= (-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-6, i)*binomial(n+k-6-2*i, n-6), i=0..(n+k)/2-6) and a(n,k)=0 if n and k are of different parity.

A373504 Triangular array: row n gives the coefficients T(n,k) of powers x^(2k) in the series expansion of ((b^n + b^(-n))/2)^2, where b = x + sqrt(x^2 + 1).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 9, 24, 16, 1, 16, 80, 128, 64, 1, 25, 200, 560, 640, 256, 1, 36, 420, 1792, 3456, 3072, 1024, 1, 49, 784, 4704, 13440, 19712, 14336, 4096, 1, 64, 1344, 10752, 42240, 90112, 106496, 65536, 16384, 1, 81, 2160, 22176, 114048, 329472, 559104, 552960, 294912, 65536

Offset: 0

Clark Kimberling, Aug 03 2024

Related to Chebyshev polynomials of the first kind; see A123588.

			First 8 rows:
  1
  1    1
  1    4     4
  1    9    24     16
  1   16    80    128     64
  1   25   200    560    640    256
  1   36   420   1792   3456   3072   1024
  1   49   784   4704  13440  19612  14336  4096
The 4th polynomial is 1 + 9 x^2 + 24 x^4 + 16 x^6.

Cf. A000012 (col 0), A000290 (col 1), A002415 ((1/4)*col(2)), A112742 (col 2), A000302 (T(n,n)), A123588, A008310.
Row sums give A055997(n+1).
Triangle without column 0 gives A334009.

p:= proc(n) option remember; (b-> series(
      ((b^n+b^(-n))/2)^2, x, 2*n+1))(x+sqrt(x^2+1))
    end:
T:= (n, k)-> coeff(p(n), x, 2*k):
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Aug 03 2024

t[n_] := ((x + Sqrt[x^2 + 1])^n + (x + Sqrt[x^2 + 1])^(-n))/2
u = Expand[Table[FullSimplify[Expand[t[n]]], {n, 0, 10}]^2]
v = Column[CoefficientList[u, x^2]] (* array *)
Flatten[v] (* sequence *)
T[n_, k_] := If[k==0, 1, 4^(k - 1)*(2*Binomial[n + k, 2*k] - Binomial[n + k -1, 2*k -1])]; Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Aug 11 2024 *)

T(n, k) = if (k=0) then 1, otherwise 4^(k - 1)*(2*binomial(n + k, 2*k) - binomial(n + k - 1, 2*k - 1)). - Detlef Meya, Aug 11 2024

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A136389 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,3}(x) with 0 omitted (exponents in increasing order).

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A136390 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,4}(x) with 0 omitted (exponents in increasing order).

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A136397 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,5}(x) with 0 omitted (exponents in increasing order).

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A136398 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,6}(x) with 0 omitted (exponents in increasing order).

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A373504 Triangular array: row n gives the coefficients T(n,k) of powers x^(2k) in the series expansion of ((b^n + b^(-n))/2)^2, where b = x + sqrt(x^2 + 1).

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