G. C. Greubel - cp's OEIS frontend

User: G. C. Greubel

G. C. Greubel has authored 84 sequences. Here are the ten most recent ones:

A364705 Expansion of 1/(1 - 4*x - x^2 + x^3).

1, 4, 17, 71, 297, 1242, 5194, 21721, 90836, 379871, 1588599, 6643431, 27782452, 116184640, 485877581, 2031912512, 8497342989, 35535406887, 148607058025, 621466295998, 2598936835130, 10868606578493, 45451896853104, 190077257155779, 794892318897727, 3324194635893583

Offset: 0

G. C. Greubel, Aug 04 2023

nonn

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

Cf. A124807, A126393.

I:=[1,4,17]; [n le 3 select I[n] else 4*Self(n-1) +Self(n-2) -Self(n-3): n in [1..41]];

LinearRecurrence[{4,1,-1}, {1,4,17}, 41]

@CachedFunction
def a(n): # a = A364705
    if (n<3): return (1,4,17)[n]
    else: return 4*a(n-1) +a(n-2) -a(n-3)
[a(n) for n in range(41)]

G.f.: 1/(1 - 4*x - x^2 + x^3).
a(n) = 4*a(n-1) + a(n-2) - a(n-3).
a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n-k, j)*binomial(n-k, k-j)*4^(n-2*k)*((1-sqrt(17))/2)^(k-j)*((1+sqrt(17))/2)^j.

A358027 Expansion of g.f.: (1 + x - 2x^2 + 2x^4)/((1-x)(1-3x^2)).

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 35, 62, 107, 188, 323, 566, 971, 1700, 2915, 5102, 8747, 15308, 26243, 45926, 78731, 137780, 236195, 413342, 708587, 1240028, 2125763, 3720086, 6377291, 11160260, 19131875, 33480782, 57395627

Offset: 0

G. C. Greubel, Oct 31 2022

Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A052993, A062318, A164123, A254006.

I:=[3,6,11]; [1,2] cat [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..60]];

LinearRecurrence[{1,3,-3}, {1,2,3,6,11}, 61]

def A254006(n): return 3^(n/2)*(1 + (-1)^n)/2
def A358027(n): return (1/3)*( 4*A254006(n) + 7*A254006(n-1) +2*int(n==0) + 2*int(n==1) - 3 )
[A358027(n) for n in (0..60)]

a(n) = (1/3)*(2*[n=0] + 2*[n=1] - 3 + 4*A254006(n) + 7*A254006(n-1)).
a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3), for n >= 5.
E.g.f.: (1/3)*( 2 + 2*x - 3*exp(x) + 4*cosh(sqrt(3)*x) + (7/sqrt(3))*sinh(sqrt(3)*x) ).
G.f.: (1 +x -2*x^2 +2*x^4)/((1-x)*(1-3*x^2)). - Clark Kimberling, Oct 31 2022

A356916 Irregular triangle A(n, q) = total number of labeled P_4 - free chordal graphs on n vertices and q edges, read by rows; also companion triangle to A058865.

Original entry on oeis.org

1, 3, 3, 1, 6, 15, 8, 12, 6, 1, 10, 45, 60, 75, 60, 80, 30, 30, 10, 1, 15, 105, 275, 420, 516, 625, 465, 540, 495, 276, 255, 80, 60, 15, 1, 21, 210, 910, 2100, 3192, 4767, 5355, 4830, 5845, 5061, 5397, 4725, 2730, 2625, 1932, 882, 630, 175, 105, 21, 1

Offset: 1

G. C. Greubel, Sep 03 2022

			The irregular triangle begins as:
   1;
   3,   3,   1;
   6,  15,   8,  12,   6,   1;
  10,  45,  60,  75,  60,  80,  30,  30,  10,   1;
  15, 105, 275, 420, 516, 625, 465, 540, 495, 276, 255, 80, 60, 15, 1;

G. C. Greubel, Rows n = 2..30 of the irregular triangle, flattened
R. Castelo and N. C. Wormald, Enumeration of P4-free chordal graphs.
R. Castelo and N. C. Wormald, Enumeration of P4-Free chordal graphs, Graphs and Combinatorics, 19:467-474, 2003.
M. C. Golumbic, Trivially perfect graphs, Discr. Math. 24(1) (1978), 105-107.
T. H. Ma and J. P. Spinrad, Cycle-free partial orders and chordal comparability graphs, Order, 1991, 8:49-61.
E. S. Wolk, A note on the comparability graph of a tree, Proc. Am. Math. Soc., 1965, 16:17-20.

Cf. A000217, A000332, A058865.

t[n_, k_]:= t[n, k]= If[k==Binomial[n, 2], 1, Sum[Binomial[n, j]*(A[n-j, k-j*(2*n -1-j)/2] - t[n-j, k-j*(2*n-1-j)/2]), {j, n-2}]]; (* t = A058865 *)
A[n_, k_]:= A[n, k]= t[n, k] + Sum[Sum[Binomial[n-1, j-1]*t[j, m]*A[n-j, k-m], {j, n-1}], {m, 0, k}]; (* A = A356916 *)
Table[A[n, k], {n,2,12}, {k,Binomial[n, 2]}]//Flatten

A356916(n, q)={A058865(n, q) + sum(k=1, n-1, k*binomial(n, k)*sum(j=k-1, k*(k-1)/2, A058865(k, j)*A356916(n-k, q-j)))/n} \\ Edited: Code for A058865 should exist and be updated only there. - M. F. Hasler, Sep 26 2022
vector(7, n, vector(binomial(n+1,2), k, A356916(n+1, k)))

@CachedFunction
def t(n,k): # t = A058865
    if (k==binomial(n,2)): return 1
    else: return sum( binomial(n,j)*( A(n-j, k-j*(2*n-1-j)/2) - t(n-j, k-j*(2*n-1-j)/2) ) for j in (1..n-2) )
@CachedFunction
def A(n,k): # A = A356916
    return t(n,k) + sum(sum( binomial(n-1,j-1)*t(j,m)*A(n-j,k-m) for j in (1..n-1) ) for m in (0..k) )
flatten([[A(n,k) for k in (1..binomial(n,2))] for n in (2..12)])

Let a(n, q) be the number of labeled connected P_4 - free chordal graphs on n vertices and q edges (see A058865), then:
a(n, q) = Sum_{k=1..n-2} binomial(n, k)*(A(n-k, q - k(2*n-1-k)/2) - a(n-k, q - k(2*n-1-k)/2)) for 1 <= q <= binomial(n,2), n >= 2, with a(n, binomial(n,2)) = 1.
A(n, q) = a(n, q) + Sum_{k = 1..n-1} binomial(n-1, k-1)*Sum_{j = k-1..min(k(k-1)/2, q)} a(k, j)*A(n-k, q-j).
A(n, binomial(n,2)) = 1, n >= 2.
A(n, 1) = A(n, binomial(n,2) - 1) = A000217(n-1), n >= 2.
A(n, 2) = 3*A000332(n+1), n >= 3.

A352972 a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k).

Original entry on oeis.org

1, 6, 35, 204, 1199, 7089, 42070, 250269, 1491262, 8896310, 53118352, 317373194, 1897253203, 11346582851, 67882263130, 406231442387, 2431626954934, 14558306758418, 87177151134954, 522110098886882, 3127380060424476, 18734897945679836, 112245303177542790, 672552484035697364, 4030148584900522009

Offset: 0

G. C. Greubel, Apr 12 2022

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A026536, A026550.

T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] +T[n-1, k-1] +T[n-1, k], T[n-1, k-2] +T[n-1, k]] ]];
A352972[n_]:= A352972[n]= Sum[T[j,k], {j,0,2*n}, {k,0,j}];
Table[A352972[n], {n,0,40}]

@CachedFunction
def T(n, k): # A026536
    if k == 0 or k == 2*n: return 1
    elif k == 1 or k == 2*n-1: return n//2
    elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
    return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
def A352972(n): return sum(sum(T(j,k) for k in (0..j)) for j in (0..2*n))
[A352972(n) for n in (3..40)]

a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k).

A334824 Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).

Original entry on oeis.org

1, 3, 0, 15, 0, -1, 105, 0, -10, 0, 945, 0, -105, 0, 1, 10395, 0, -1260, 0, 21, 0, 135135, 0, -17325, 0, 378, 0, -1, 2027025, 0, -270270, 0, 6930, 0, -36, 0, 34459425, 0, -4729725, 0, 135135, 0, -990, 0, 1, 654729075, 0, -91891800, 0, 2837835, 0, -25740, 0, 55, 0, 13749310575, 0, -1964187225, 0, 64324260, 0, -675675, 0, 2145, 0, -1

Offset: 0

G. C. Greubel, May 13 2020, following a suggestion from Michel Marcus

Lambert's denominator polynomials related to convergents of tan(x), f(n, x), are given in A334823.

			Polynomials:
g(0, x) = 1;
g(1, x) = 3*x;
g(2, x) = 15*x^2 - 1;
g(3, x) = 105*x^3 - 10*x;
g(4, x) = 945*x^4 - 105*x^2 + 1;
g(5, x) = 10395*x^5 - 1260*x^3 + 21*x;
g(6, x) = 135135*x^6 - 17325*x^4 + 378*x^2 - 1;
g(7, x) = 2027025*x^7 - 270270*x^5 + 6930*x^3 - 36*x.
Triangle of coefficients begins as:
        1;
        3, 0;
       15, 0,      -1;
      105, 0,     -10, 0;
      945, 0,    -105, 0,    1;
    10395, 0,   -1260, 0,   21, 0;
   135135, 0,  -17325, 0,  378, 0,  -1;
  2027025, 0, -270270, 0, 6930, 0, -36, 0.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
J.-H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendantes et logarithmiques (Memoir on some properties that can be traced from circular transcendent and logarithmic quantities), Histoire de l’Académie royale des sciences et belles-lettres (1761), Berlin. See also.

Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094675, A334823.

Columns k: A001147 (k=0), A000457 (k=2), A001881 (k=4), A130563 (k=6).

C := ComplexField();
T:= func< n, k| Round( i^k*Factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k+1)*Factorial(n-k)) ) >;
[T(n,k): k in [0..n], n in [0..10]];

Maple

T:= (n, k) -> I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!); seq(seq(T(n, k), k = 0..n), n = 0..10);

Mathematica

(* First program *) y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x]; g[n_, k_]:= Coefficient[((-I)^n/2)*(y[n+1, I*x] + (-1)^n*y[n+1, -I*x]), x, k]; Table[g[n, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Second program *) Table[I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten

Sage

[[ i^k*factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*factorial(k+1)*factorial(n-k)) for k in (0..n)] for n in (0..10)]

Formula

Equals the coefficients of the polynomials, g(n, x), defined by: (Start)

g(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k+1)!/((2*k+1)!*(n-2*k)!))*(x/2)^(n-2*k).

g(n, x) = ((2*n+1)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 3/2, -n, -n-1/2; -1/x^2).

g(n, x) = ((-i)^n/2)*(y(n+1, i*x) + (-1)^n*y(n+1, -i*x)), where y(n, x) are the Bessel Polynomials.

g(n, x) = (2*n-1)*x*g(n-1, x) - g(n-2, x).

E.g.f. of g(n, x): sin((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).

g(n, 1) = (-1)^n*g(n, -1) = A053984(n) = (-1)^n*A053983(-n-1) = (-1)^n*f(-n-1, 1).

g(n, 2) = (-1)^n*g(n, -2) = A053987(n+1). (End)

As a number triangle:

T(n, k) = i^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), where i = sqrt(-1).

T(n, 0) = A001147(n+1).

A334823 Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).

Original entry on oeis.org

1, 1, 0, 3, 0, -1, 15, 0, -6, 0, 105, 0, -45, 0, 1, 945, 0, -420, 0, 15, 0, 10395, 0, -4725, 0, 210, 0, -1, 135135, 0, -62370, 0, 3150, 0, -28, 0, 2027025, 0, -945945, 0, 51975, 0, -630, 0, 1, 34459425, 0, -16216200, 0, 945945, 0, -13860, 0, 45, 0, 654729075, 0, -310134825, 0, 18918900, 0, -315315, 0, 1485, 0, -1

Offset: 0

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Author

G. C. Greubel, May 12 2020, following a suggestion from Michel Marcus

Keywords

tabl
sign

Comments

Lambert's numerator polynomials related to convergents of tan(x), g(n, x), are given in A334824.

Examples

Polynomials: f(0, x) = 1; f(1, x) = x; f(2, x) = 3*x^2 - 1; f(3, x) = 15*x^3 - 6*x; f(4, x) = 105*x^4 - 45*x^2 + 1; f(5, x) = 945*x^5 - 420*x^3 + 15*x; f(6, x) = 10395*x^6 - 4725*x^4 + 210*x^2 - 1; f(7, x) = 135135*x^7 - 62370*x^5 + 3150*x^3 - 28*x; f(8, x) = 2027025*x^8 - 945945*x^6 + 51975*x^4 - 630*x^2 + 1. Triangle of coefficients begins as: 1; 1, 0; 3, 0, -1; 15, 0, -6, 0; 105, 0, -45, 0, 1; 945, 0, -420, 0, 15, 0; 10395, 0, -4725, 0, 210, 0, -1; 135135, 0, -62370, 0, 3150, 0, -28, 0; 2027025, 0, -945945, 0, 51975, 0, -630, 0, 1.

Links

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

J.-H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendantes et logarithmiques (Memoir on some properties that can be traced from circular transcendent and logarithmic quantities), Histoire de l’Académie royale des sciences et belles-lettres (1761), Berlin. See also.

Crossrefs

Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094674, A334824.

Columns k: A001147 (k=0), A001879 (k=2), A001880 (k=4), A038121 (k=6).

Programs

Magma

C := ComplexField(); T:= func< n, k| Round( i^k*Factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k)*Factorial(n-k)) ) >; [T(n,k): k in [0..n], n in [0..10]];

Maple

T:= (n, k) -> I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!); seq(seq(T(n, k), k = 0 .. n), n = 0 .. 10);

Mathematica

(* First program *) y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x]; f[n_, k_]:= Coefficient[((-I)^n/2)*(y[n, I*x] + (-1)^n*y[n, -I*x]), x, k]; Table[f[n, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Second program *) Table[ I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten

Sage

[[ i^k*factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*factorial(k)*factorial(n-k)) for k in (0..n)] for n in (0..10)]

Formula

Equals the coefficients of the polynomials, f(n, x), defined by: (Start)

f(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k)!/((2*k)!*(n-2*k)!))*(x/2)^(n-2*k).

f(n, x) = ((2*n)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 1/2, -n, -n+1/2; -1/x^2).

f(n, x) = ((-i)^n/2)*(y(n, i*x) + (-1)^n*y(n, -i*x)), where y(n, x) are the Bessel Polynomials.

f(n, x) = (2*n-1)*x*f(n-1, x) - f(n-2, x).

E.g.f. of f(n, x): cos((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).

f(n, 1) = (-1)^n*f(n, -1) = A053983(n) = (-1)^(n+1)*A053984(-n-1) = (-1)^(n+1) * g(-n-1, 1).

f(n, 2) = (-1)^n*f(n, -2) = A053988(n+1). (End)

As a number triangle:

T(n, k) = i^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), where i = sqrt(-1).

T(n, 0) = A001147(n).

A330767 a(n) = 25*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 25.

Original entry on oeis.org

2, 25, 627, 15700, 393127, 9843875, 246490002, 6172093925, 154548838127, 3869893047100, 96901875015627, 2426416768437775, 60757321085960002, 1521359443917437825, 38094743419021905627, 953889944919465078500, 23885343366405648868127, 598087474105060686781675, 14976072195992922818410002

Offset: 0

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Author

G. C. Greubel, Dec 29 2019

Keywords

nonn
easy

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (25,1).

Crossrefs

Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), this sequence (m=25).

Programs

GAP

a:=[2,25];; for n in [3..25] do a[n]:=25*a[n-1]+a[n-2]; od; a;

Magma

I:=[2,25]; [n le 2 select I[n] else 25*Self(n-1) +Self(n-2): n in [1..25]];

Maple

seq(simplify(2*(-I)^n*ChebyshevT(n, 25*I/2)), n = 0..25);

Mathematica

LucasL[Range[25] -1, 25]

PARI

vector(26, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 25*I/2) )

Sage

[2*(-I)^n*chebyshev_T(n, 25*I/2) for n in (0..25)]

Formula

a(n) = ( (25 + sqrt(629))^n + (25 - sqrt(629))^n )/2^n.

G.f.: (2 - 25*x)/(1-25*x-x^2).

a(n) = Lucas(n, 25) = 2*(-i)^n * ChebyshevT(n, 25*i/2).

A328141 a(n) = a(n-1) - (n-2)*a(n-2), with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 2, 0, -4, -4, 12, 32, -40, -264, 56, 2432, 1872, -24880, -47344, 276096, 938912, -3202528, -18225120, 36217856, 364270016, -323869248, -7609269568, -808015360, 166595915136, 185180268416, -3813121694848, -8442628405248, 90698535660800, 318649502602496, -2220909495899904

Offset: 0

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G. C. Greubel, Oct 04 2019

Keywords

sign

Comments

Former title and formula of A122033, but not the data.

Links

G. C. Greubel, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A001464, A060821, A121966, A122033.

Programs

GAP

a:=[1,2];; for n in [3..35] do a[n]:=a[n-1]-(n-3)*a[n-2]; od; a;

Magma

I:=[1,2]; [n le 2 select I[n] else Self(n-1) - (n-3)*Self(n-2): n in [1..35]];

Maple

a:= proc (n) option remember; if n < 2 then n+1 else a(n-1) - (n-2)*a(n-2) fi; end proc; seq(a(n), n = 0..35);

Mathematica

a[n_]:= a[n]= If[n<2, n+1, a[n-1]-(n-2)*a[n-2]]; Table[a[n], {n,0,35}]

PARI

my(m=35, v=concat([1,2], vector(m-2))); for(n=3, m, v[n] = v[n-1] - (n-3)*v[n-2] ); v

Sage

def a(n): if n<2: return n+1 else: return a(n-1) - (n-2)*a(n-2) [a(n) for n in (0..35)]

Formula

a(n) = a(n-1) - (n-2)*a(n-2), with a(0)=1, a(1)=2.

E.g.f.: 1 + sqrt(2*e*Pi)*( erf(1/sqrt(2)) + erf((x-1)/sqrt(2)) ), where erf(x) is the error function.

a(n) = 2*(-1)^(n-1)*A001464(n-1).

a(n) = 2*(1/sqrt(2))^(n-1) * Hermite(n-1, 1/sqrt(2)), n > 0.

A306547 Triangle read by rows, defined by Riordan's general Eulerian recursion: T(n, k) = (k+3)*T(n-1, k) + (n-k-2) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).

Original entry on oeis.org

1, 1, -2, 1, -11, 4, 1, -55, 35, -8, 1, -274, 210, -91, 16, 1, -1368, 986, -637, 219, -32, 1, -6837, 3180, -3473, 1752, -507, 64, 1, -34181, -1431, -17951, 10543, -4563, 1147, -128, 1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256, 1, -854494, -1726360, -1490890, -2314, -177832, 84176, -28105, 5627, -512

Offset: 1

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G. C. Greubel, Feb 22 2019

Keywords

sign
tabl

Comments

Row sums are {1, -1, -6, -27, -138, -831, -5820, -46563, -419070, -4190703, ...}.

The Mathematica code for e(n,k,m) gives eleven sequences of which the first few are in the OEIS (see Crossrefs section).

Examples

Triangle begins with: 1. 1, -2. 1, -11, 4. 1, -55, 35, -8. 1, -274, 210, -91, 16. 1, -1368, 986, -637, 219, -32. 1, -6837, 3180, -3473, 1752, -507, 64. 1, -34181, -1431, -17951, 10543, -4563, 1147, -128. 1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256.

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215.

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3), this sequence.

Programs

Mathematica

e[n_, 0, m_]:= 1; (* Example for m=3 *) e[n_, k_, m_]:= 0 /; k >= n; e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m]; Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}] T[n_, 1]:= 1; T[n_, n_]:= (-2)^(n-1); T[n_, k_]:= T[n, k] = (k+3)*T[n-1, k] + (n-k-2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]//Flatten

PARI

{T(n, k) = if(k==1, 1, if(k==n, (-2)^(n-1), (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)))}; for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")))

Sage

def T(n, k): if (k==1): return 1 elif (k==n): return (-2)^(n-1) else: return (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1) [[T(n, k) for k in (1..n)] for n in (1..12)]

Formula

T(n, k) = (k+3)*T(n-1, k) + (n-k-2)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).

e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.

A306183 The coefficients of x in the reduction of x^2 -> x + 1 for the polynomial p(n,x) = Product_{k=1..n} (x+k).

Original entry on oeis.org

0, 1, 4, 19, 108, 719, 5496, 47465, 457160, 4858865, 56490060, 713165035, 9715762980, 142069257055, 2219386098160, 36889108220305, 650018185589520, 12103669982341025, 237476572759473300, 4896758300881695875, 105866710959427454300, 2394660132226522508975, 56560492065670933962600

Offset: 0

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G. C. Greubel, Feb 07 2019

Keywords

nonn

Comments

See A192936 for the constant term of the reduction x^2 -> x + 1 for the polynomial p(n,x) = Product_{k=1..n} (x+k).

Links

G. C. Greubel, Table of n, a(n) for n = 0..445

Crossrefs

Cf. A192936, A323620 (signed)

Programs

Magma

[(-1)^(n+1)*(&+[StirlingFirst(n+2,k)*Fibonacci(k): k in [0..n+2]]): n in [0..30]];

Mathematica

Table[(-1)^(n+1)*Sum[StirlingS1[n+2,k]*Fibonacci[k],{k,0,n+2}],{n,0,30}]

PARI

{a(n) = (-1)^(n+1)*sum(k=0,n+2, stirling(n+2,k,1)*fibonacci(k))}; vector(30, n, n--; a(n))

Sage

[sum((-1)^(k+1)*stirling_number1(n+2,k)*fibonacci(k) for k in (0..n+2)) for n in (0..30)]

Formula

a(n) = (-1)^(n+1)*Sum_{k=0..n+2} Stirling1(n+2,k)*A000045(k).

From Vaclav Kotesovec, Feb 09 2019: (Start)

a(n) = 2*n*a(n-1) - (n^2 - n - 1)*a(n-2).

a(n) = cos(Pi*sqrt(5)/2) * (Gamma(sqrt(5)*phi) * Gamma(n + 1/phi^2) / phi^2 - phi^2 * Gamma(sqrt(5)/phi) * Gamma(n + phi^2)) / (Pi*sqrt(5)).

a(n) ~ c * n! * n^phi, where c = -cos(sqrt(5)*Pi/2) * (5 + 3*sqrt(5)) * Gamma((5 - sqrt(5))/2) / (10*Pi) = 0.30858712435869... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. (End)

cp's OEIS Frontend

User: G. C. Greubel

A364705 Expansion of 1/(1 - 4*x - x^2 + x^3).

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A358027 Expansion of g.f.: (1 + x - 2*x^2 + 2*x^4)/((1-x)*(1-3*x^2)).

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A356916 Irregular triangle A(n, q) = total number of labeled P_4 - free chordal graphs on n vertices and q edges, read by rows; also companion triangle to A058865.

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A352972 a(n) = Sum_{j=0..2*n} Sum_{k=0..j} A026536(j, k).

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A334824 Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).

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Maple

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A334823 Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).

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Maple

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A330767 a(n) = 25*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 25.

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A358027 Expansion of g.f.: (1 + x - 2x^2 + 2x^4)/((1-x)(1-3x^2)).

A306547 Triangle read by rows, defined by Riordan's general Eulerian recursion: T(n, k) = (k+3)T(n-1, k) + (n-k-2) T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).