A002700 -id:A002700 - cp's OEIS frontend

A053125 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in decreasing order).

1, 4, -2, 16, -16, 3, 64, -96, 40, -4, 256, -512, 336, -80, 5, 1024, -2560, 2304, -896, 140, -6, 4096, -12288, 14080, -7680, 2016, -224, 7, 16384, -57344, 79872, -56320, 21120, -4032, 336, -8, 65536, -262144, 430080, -372736, 183040, -50688, 7392, -480, 9, 262144, -1179648, 2228224, -2293760, 1397760

Offset: 0

Wolfdieter Lang

A000302 (powers of 4), A002699, A002700 unsigned column sequences for m=0..2.
G.f. for row polynomials U(n,2*x-1) and row sums same as for A053124.
With offset 1 this is also the coefficient triangle of 2* U(2*n-1,x) expanded in decreasing powers of x. W. Lang, Mar 07 2007.

			{1}; {4,-2}; {16,-16,3}; {64,-96,40,-4}; {256,-512,336,-80,5};... E.g. fourth row (n=3) corresponds to polynomial U^{*}(3,m)=U(3,2*x-1)= 64*x^3-96*x^2+40*x-4.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Rows n=0..50 of triangle, flattened
W. Lang, First rows and related triangles .
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Index entries for sequences related to Chebyshev polynomials.

Cf. A053124, A053123.

Reverse /@ CoefficientList[Table[ChebyshevU[n, 2 x - 1], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)
Reverse /@ CoefficientList[ChebyshevU[Range[0, 10], 2 x - 1], x] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)

a(n, m) = A053124(n, n-m)= (4^(n-m))*A053123(n, m)= (4^(n-m))*((-1)^m)*binomial(2*n+1-m, m) if n >= m, else 0.

a(n, m) := -2*a(n-1, m-1)+4*a(n-1, m)-a(n-2, m-2), a(-2, m) := 0=: a(n, -2), a(-1, m) := 0=: a(n, -1), a(0, 0)=1, a(n, m)=0 if n

G.f. for m-th column (signed triangle): ((-x)^m)*Po(m+1, 4*x)/(1-4*x)^(m+1), with Po(k, x) := sum('binomial(k, 2*j+1)*x^j', 'j'=0..floor(k/2)).

A229694 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

0, 0, 0, 1, 3, 0, 3, 43, 40, 0, 12, 245, 626, 336, 0, 40, 1171, 5077, 6732, 2304, 0, 120, 5077, 35825, 80757, 62856, 14080, 0, 336, 20691, 230383, 848937, 1125333, 539568, 79872, 0, 896, 80757, 1400413, 8186713, 17724789, 14461173, 4377888, 430080, 0

Offset: 1

R. H. Hardin, Sep 27 2013

			Some solutions for n=3, k=4:
  0 1 2 1     0 1 0 2     0 1 0 2     0 0 1 2     0 1 0 2
  0 1 0 2     0 2 0 2     0 2 1 0     1 0 0 1     0 2 1 1
  1 2 1 1     2 1 2 0     2 2 1 2     2 1 2 0     1 0 2 2
Table starts
.0......0........1..........3...........12............40.............120
.0......3.......43........245.........1171..........5077...........20691
.0.....40......626.......5077........35825........230383.........1400413
.0....336.....6732......80757.......848937.......8186713........75035643
.0...2304....62856....1125333.....17724789.....258006388......3583403667
.0..14080...539568...14461173....342532665....7551515197....159377253183
.0..79872..4377888..175867605...6279934941..210095323918...6749642728251
.0.430080.34105536.2054728053.110801828529.5632122625852.275739382892979

R. H. Hardin, Table of n, a(n) for n = 1..286

Column 2 is A002700(n+1).

Row 1 is A052482(n-2).

Empirical for column k:
k=1: a(n) = a(n-1).
k=2: a(n) = 12*a(n-1) - 48*a(n-2) + 64*a(n-3).
k=3: a(n) = 18*a(n-1) - 108*a(n-2) + 216*a(n-3) for n > 4.
k=4: a(n) = 27*a(n-1) - 243*a(n-2) + 729*a(n-3) for n > 4.
k=5: [order 6] for n > 7.
k=6: [order 9] for n > 11.
k=7: [order 12] for n > 14.
Empirical for row n:
n=1: a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n > 6.
n=2: a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) for n > 6.
n=3: a(n) = 15*a(n-1) - 81*a(n-2) + 185*a(n-3) - 162*a(n-4) + 60*a(n-5) - 8*a(n-6) for n > 10.
n=4: [order 9] for n > 17.
n=5: [order 21] for n > 27.
n=6: [order 29] for n > 39.
n=7: [order 86] for n > 94.

A225467 Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 3, 4, 9, 40, 16, 27, 316, 336, 64, 81, 2320, 4960, 2304, 256, 243, 16564, 63840, 54400, 14080, 1024, 729, 116920, 768496, 1071360, 485120, 79872, 4096, 2187, 821356, 8921136, 19144384, 13502720, 3777536, 430080, 16384, 6561, 5758240, 101417920, 322850304

Offset: 0

Peter Luschny, May 08 2013

The definition of the Stirling-Frobenius subset numbers of order m is in A225468.

This is the Sheffer triangle (exp(3*x), exp(4*x) - 1). See also the P. Bala link under A225469, the Sheffer triangle (exp(3*x),(1/4)*(exp(4*x) - 1)), which is named there exponential Riordan array S_{(4,0,3)}. - Wolfdieter Lang, Apr 13 2017

			[n\k][  0,      1,       2,        3,        4,       5,      6,     7]
[0]     1,
[1]     3,      4,
[2]     9,     40,      16,
[3]    27,    316,     336,       64,
[4]    81,   2320,    4960,     2304,      256,
[5]   243,  16564,   63840,    54400,    14080,    1024,
[6]   729, 116920,  768496,  1071360,   485120,   79872,   4096,
[7]  2187, 821356, 8921136, 19144384, 13502720, 3777536, 430080, 16384.
...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence (see the Maple program): T(4, 2) = 4*T(3, 1) + (4*2+3)*T(3, 2) = 4*316 + 11*336 = 4960.
Boas-Buck recurrence for column k = 2, and n = 4: T(4, 2) = (1/2)*(2*(6 + 4*2)*T(3, 2) + 2*6*(-4)^2*Bernoulli(2)*T(2, 2)) = (1/2)*(28*336 + 12*16*(1/6)*16) = 4960. (End)

Vincenzo Librandi, Rows n = 0..50, flattened
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 9.
Peter Luschny, Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A048993 (m=1), A154537 (m=2), A225466 (m=3). A225469 (scaled).

Cf. Columns: A000244, 4*A016138, 16*A018054. A225118.

SF_SS := proc(n, k, m) option remember;
if n = 0 and k = 0 then return(1) fi;
if k > n or  k < 0 then return(0) fi;
m*SF_SS(n-1, k-1, m) + (m*(k+1)-1)*SF_SS(n-1, k, m) end:
seq(print(seq(SF_SS(n, k, 4), k=0..n)), n=0..5);

EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = (If[ n == 0, Return[If[k == 0, 1, 0]]]; Return[(m*(n-k)+m-1)*EulerianNumber[n-1, k-1, m] + (m*k+1)*EulerianNumber[n-1, k, m]]); SFSS[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/k!; Table[ SFSS[n, k, 4], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2013, translated from Sage *)

T(n, k) = sum(m=0, k, binomial(k, m)*(-1)^(m - k)*((3 + 4*m)^n)/k!);
for(n = 0, 10, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 13 2017

from sympy import binomial, factorial
def T(n, k): return sum(binomial(k, m)*(-1)**(m - k)*((3 + 4*m)**n)//factorial(k) for m in range(k + 1))
for n in range(11): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 13 2017

@CachedFunction
def EulerianNumber(n, k, m) :
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1,k-1,m)+(m*k+1)*EulerianNumber(n-1,k,m)
def SF_SS(n, k, m):
    return add(EulerianNumber(n,j,m)*binomial(j,n-k) for j in (0..n))/factorial(k)
def A225467(n): return SF_SS(n, k, 4)

T(n, k) = (1/k!)*sum_{j=0..n} binomial(j, n-k)*A_4(n, j) where A_m(n, j) are the generalized Eulerian numbers A225118.
For a recurrence see the Maple program.
T(n, 0) ~ A000244; T(n, 1) ~ A190541.
T(n, n) ~ A000302; T(n, n-1) ~ A002700.
From Wolfdieter Lang, Apr 13 2017: (Start)
T(n, k) = Sum_{m=0..k} binomial(k,m)*(-1)^(m-k)*((3+4*m)^n)/k!, 0 <= k <= n.
In terms of Stirling2 = A048993: T(n, m) = Sum_{k=0..n} binomial(n, k)* 3^(n-k)*4^k*Stirling2(k, m), 0 <= m <= n.
E.g.f. exp(3*z)*exp(x*(exp(4*z) - 1)) (Sheffer property).
E.g.f. column k: exp(3*x)*((exp(4*x) - 1)^k)/k!, k >= 0.
O.g.f. column k: (4*x)^k/Product_{j=0..k} (1 - (3 + 4*j)*x), k >= 0.
(End)
Boas-Buck recurrence for column sequence k: T(n, k) = (1/(n - k))*((n/2)*(6 + 4*k)*T(n-1, k) + k*Sum_{p=k..n-2} binomial(n, p)*(-4)^(n-p)*Bernoulli(n-p)*T(p, k)), for n > k >= 0, with input T(k, k) = 4^k. See a comment and references in A282629. An example is given below. - Wolfdieter Lang, Aug 11 2017

A054322 Fourth unsigned column of Lanczos triangle A053125 (decreasing powers).

Original entry on oeis.org

4, 80, 896, 7680, 56320, 372736, 2293760, 13369344, 74711040, 403701760, 2122317824, 10905190400, 54962159616, 272461987840, 1331439861760, 6425271074816, 30666066493440, 144929376436224, 678948430151680

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Cf. A053125, A053123, A002700, A054329.

List([0..20], n-> 4^n*Binomial(2*n+4, 3)); # G. C. Greubel, Jul 22 2019

[4^n*Binomial(2*n+4, 3): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[4^n*Binomial[2*n+4, 3], {n,0,20}] (* G. C. Greubel, Jul 22 2019 *)
LinearRecurrence[{16,-96,256,-256},{4,80,896,7680},20] (* Harvey P. Dale, Mar 27 2023 *)

vector(20, n, n--; 4^n*binomial(2*n+4, 3)) \\ G. C. Greubel, Jul 22 2019

[4^n*binomial(2*n+4, 3) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 4^n*binomial(2*n+4, 3) = -A053125(n+3, 3) = 4*A054329(n).
G.f.: 4*(1+4*x)/(1-4*x)^4.
E.g.f.: (4/3)*(3 +48*x +120*x^2 +64*x^3)*exp(4*x). - G. C. Greubel, Jul 22 2019

A225478 Triangle read by rows, 4^k*s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 3, 4, 21, 40, 16, 231, 524, 336, 64, 3465, 8784, 7136, 2304, 256, 65835, 180756, 170720, 72320, 14080, 1024, 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096, 40883535, 125416476, 143221680, 81946816, 25939200, 4609024, 430080, 16384, 1267389585, 4051444896, 4941537984, 3113238016, 1131902464, 246636544, 31768576, 2228224, 65536

Offset: 0

Peter Luschny, May 17 2013

Triangle T(n,k), read by rows, given by (3, 4, 7, 8, 11, 12, 15, 16, ... (A014601)) DELTA (4, 0, 4, 0, 4, 0, 4, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 14 2015.

			[n\k][    0,       1,       2,       3,      4,     5,    6 ]
[0]       1,
[1]       3,       4,
[2]      21,      40,      16,
[3]     231,     524,     336,      64,
[4]    3465,    8784,    7136,    2304,    256,
[5]   65835,  180756,  170720,   72320,  14080,  1024,
[6] 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096.

Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

T(n, 0) ~ A008545; T(n, n) ~ A000302; T(n, n-1) ~ A002700.
row sums ~ A034176; alternating row sums ~ A008545.
Cf. A225471, A132393 (m=1), A028338 (m=2), A225477 (m=3).

s[][0, 0] = 1; s[m][n_, k_] /; (k > n || k < 0) = 0; s[m_][n_, k_] := s[m][n, k] = s[m][n - 1, k - 1] + (m*n - 1)*s[m][n - 1, k];
T[n_, k_] := 4^k*s[4][n, k];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

@CachedFunction
def SF_CS(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return m*SF_CS(n-1, k-1, m) + (m*n-1)*SF_CS(n-1, k, m)
for n in (0..8): [SF_CS(n, k, 4) for k in (0..n)]

For a recurrence see the Sage program.

T(n,k) = 4^k * A225471(n,k). - Philippe Deléham, May 14 2015.

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A053125 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in decreasing order).

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A229694 T(n,k) = number of defective 3-colorings of an n X k 0..2 array connected horizontally and antidiagonally with exactly two mistakes, and colors introduced in row-major 0..2 order.

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A225467 Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

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A054322 Fourth unsigned column of Lanczos triangle A053125 (decreasing powers).

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A225478 Triangle read by rows, 4^k*s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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