A084775 -id:A084775 - cp's OEIS frontend

A084776 a(n) = sum of absolute-valued coefficients of (1+2*x-x^2)^n.

1, 4, 12, 36, 100, 300, 776, 2412, 6304, 19036, 50952, 148896, 393452, 1211444, 3167004, 9672772, 25295248, 76084796, 200590424, 608621376, 1617201648, 4908511140, 12658776540, 38907904188, 102775961200, 310485090044

Offset: 0

Paul D. Hanna, Jun 14 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A084775, A084777, A084778, A084779, A084780.

m:=40;
R:=PowerSeriesRing(Integers(), 2*(m+2));
f:= func< n,k | Coefficient(R!( (1+2*x-x^2)^n ), k) >;
[(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // G. C. Greubel, Jun 03 2023

T[n_, k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-x^2)^n,{x,0,2n}], k];
a[n_]:= a[n]= Sum[Abs[T[n,k]], {k,0,2n}];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jun 03 2023 *)

for(n=0,40,S=sum(k=0,2*n,abs(polcoeff((1+2*x-x^2)^n,k,x))); print1(S","))

def f(n,k):
    P. = PowerSeriesRing(QQ)
    return P( (1+2*x-x^2)^n ).list()[k]
def a(n): return sum( abs(f(n,k)) for k in range(2*n+1) )
[a(n) for n in range(41)] # G. C. Greubel, Jun 03 2023

a(n) = Sum_{k=0..2*n} abs(f(n, k)), where f(n, k) = (sqrt(2) - 1)^k * Sum_{j=0..k} binomial(n, j)*binomial(n, k-j)*(-1)^j*(1+sqrt(2))^(2*j). - G. C. Greubel, Jun 03 2023

A084777 a(n) = sum of absolute-valued coefficients of (1+2x-2x^2)^n.

Original entry on oeis.org

1, 5, 17, 73, 273, 881, 3785, 13081, 48737, 184321, 632193, 2526305, 8854081, 32077921, 124093025, 428178641, 1638563969, 5878561921, 21469361537, 82252171393, 286863949025, 1061000856417, 3998983314849, 14361380710817

Offset: 0

Paul D. Hanna, Jun 14 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A084775, A084776, A084778, A084779, A084780.

m:=40;
R:=PowerSeriesRing(Integers(), 2*(m+2));
f:= func< n,k | Coefficient(R!( (1+2*x-2*x^2)^n ), k) >;
[(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // G. C. Greubel, Jun 03 2023

T[n_,k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-2*x^2)^n,{x,0,2n}],k];
a[n_]:= a[n]= Sum[Abs[T[n,k]], {k,0,2n}];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jun 03 2023 *)

for(n=0,40,S=sum(k=0,2*n,abs(polcoeff((1+2*x-2*x^2)^n,k,x))); print1(S","))

def f(n,k):
    P. = PowerSeriesRing(QQ)
    return P( (1+2*x-2*x^2)^n ).list()[k]
def a(n): return sum( abs(f(n,k)) for k in range(2*n+1) )
[a(n) for n in range(41)] # G. C. Greubel, Jun 03 2023

a(n) = Sum_{k=0..2*n} abs(f(n, k)), where f(n, k) = ((sqrt(3) -1)/2)^k * Sum_{j=0..k} binomial(n, j)*binomial(n, k-j)*(-1)^j*((1+sqrt(3))/2 )^(2*j). - G. C. Greubel, Jun 03 2023

A084778 a(n) = sum of absolute-valued coefficients of (1+2x-3x^2)^n.

Original entry on oeis.org

1, 6, 28, 128, 660, 3016, 13108, 64112, 304068, 1332992, 6514356, 29341384, 131904528, 623547112, 2990903464, 13436119424, 61647598484, 284398511848, 1302463169256, 6195158123688, 28653898573420, 130138400720504

Offset: 0

Paul D. Hanna, Jun 14 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A084775, A084776, A084777, A084779, A084780.

m:=40;
R:=PowerSeriesRing(Integers(), 2*(m+2));
f:= func< n,k | Coefficient(R!( (1+2*x-3*x^2)^n ), k) >;
[(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // G. C. Greubel, Jun 04 2023

T[n_,k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-3*x^2)^n,{x,0,2n}], k];
a[n_]:= a[n]= Sum[Abs[T[n,k]], {k,0,2n}];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jun 04 2023 *)

for(n=0,40,S=sum(k=0,2*n,abs(polcoeff((1+2*x-3*x^2)^n,k,x))); print1(S","))

def f(n,k):
    P. = PowerSeriesRing(QQ)
    return P( (1+2*x-3*x^2)^n ).list()[k]
def a(n): return sum( abs(f(n,k)) for k in range(2*n+1) )
[a(n) for n in range(41)] # G. C. Greubel, Jun 04 2023

a(n) = Sum_{k=0..2*n} abs(f(n,k)), where f(n, k) = Sum_{j=0..k} binomial(n, j)*binomial(n, k-j)*(-3)^j = binomial(n, k)*Hypergeometric2F1([-n, -k], [n-k+1], -3) = (n!/k!)*4^n*(-3)^((k-n)/2)*LegendreP(n, k-n, -1/2). - G. C. Greubel, Jun 04 2023

A084779 a(n) = sum of absolute-valued coefficients of (1+2x-4x^2)^n.

Original entry on oeis.org

1, 7, 41, 207, 1313, 7807, 42593, 232463, 1290433, 7604415, 42034721, 236031231, 1363681121, 7457831007, 39670144513, 231087069823, 1291433872385, 7373001299199, 41437235793921, 229538650588863, 1268719471103233

Offset: 0

Paul D. Hanna, Jun 14 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A084775, A084776, A084777, A084778, A084780.

m:=40;
R:=PowerSeriesRing(Integers(), 2*(m+2));
f:= func< n,k | Coefficient(R!( (1+2*x-4*x^2)^n ), k) >;
[(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // G. C. Greubel, Jun 04 2023

T[n_,k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-4*x^2)^n,{x,0,2n}],k];
a[n_]:= a[n]= Sum[Abs[T[n,k]], {k,0,2n}];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jun 04 2023 *)

for(n=0,40,S=sum(k=0,2*n,abs(polcoeff((1+2*x-4*x^2)^n,k,x))); print1(S","))

def f(n,k):
    P. = PowerSeriesRing(QQ)
    return P( (1+2*x-4*x^2)^n ).list()[k]
def a(n): return sum( abs(f(n,k)) for k in range(2*n+1) )
[a(n) for n in range(41)] # G. C. Greubel, Jun 04 2023

a(n) = Sum_{k=0..2*n} abs(f(n, k)), where f(n, k) = (n!/(2*n-k)!)*i*(k-n)*2^k*5^(n/2)*LegendreP(n, n-k, 1/sqrt(5)). - G. C. Greubel, Jun 04 2023

A084780 a(n) = sum of absolute-valued coefficients of (1+3*x-x^2)^n.

Original entry on oeis.org

1, 5, 21, 77, 291, 1119, 3523, 15007, 50923, 182669, 701121, 2379129, 8909361, 32490021, 106309861, 423990203, 1456199483, 5089398187, 19942506259, 65753622619, 252337832801, 903751067081, 3026099773993, 11771846189609

Offset: 0

Paul D. Hanna, Jun 14 2003

nonn

The expansion of (1 + a*x - b*x^2)^n is: (1 + a*x - b*x^2)^n = Sum_{k=0..2*n} f(n, k)*x^k, where f(n, k) = (n!/(2*n-k)!) * (-b)^((k-n)/2) * (a^2 + 4*b)^(n/2) * LegendreP(n, n-k, a/sqrt(a^2 + 4*b)). - G. C. Greubel, Jun 04 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A084775, A084776, A084777, A084778, A084779.

m:=40;
R:=PowerSeriesRing(Integers(), 2*(m+2));
f:= func< n,k | Coefficient(R!( (1+3*x-x^2)^n ), k) >;
[(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // G. C. Greubel, Jun 04 2023

Table[Total[Abs[CoefficientList[Expand[(1+3x-x^2)^n],x]]],{n,0,30}] (* Harvey P. Dale, Jan 04 2012 *)

for(n=0,40,S=sum(k=0,2*n,abs(polcoeff((1+3*x-x^2)^n,k,x))); print1(S","))

def f(n,k):
    P. = PowerSeriesRing(QQ)
    return P( (1+3*x-x^2)^n ).list()[k]
def a(n): return sum( abs(f(n,k)) for k in range(2*n+1) )
[a(n) for n in range(41)] # G. C. Greubel, Jun 04 2023

a(n) = Sum_{k=0..2*n} abs(f(n, k)), where f(n, k) = (n!/(2*n-k)!) * i^(k-n)*(13)^(n/2)*LegendreP(n, n-k, 3/sqrt(13)).. - G. C. Greubel, Jun 04 2023

A192205 a(n) = sum of absolute values of coefficients in (1-x-x^2+x^3)^n.

Original entry on oeis.org

1, 4, 12, 36, 116, 344, 1104, 3280, 10456, 31152, 98804, 295988, 935876, 2811540, 8870324, 26695724, 84060148, 253376840, 796635360, 2404558304, 7549431884, 22820942416, 71541295984, 216562743948, 677938097756, 2054922521644

Offset: 0

Paul D. Hanna, Jun 25 2011

nonn

Conjecture: limit a(n)^(1/n) = 16*sqrt(3)/9 = 3.079201..., which is substantiated by the observation that the sums of the coefficients squared in (1-x-x^2+x^3)^n equals binomial(4n,n) (cf. A005810).

			The triangle A227964 of coefficients in (1+x-x^2-x^3)^n begins:
n=0: [1];
n=1: [1, -1, -1, 1];
n=2: [1, -2, -1, 4, -1, -2, 1];
n=3: [1, -3, 0, 8, -6, -6, 8, 0, -3, 1];
n=4: [1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1];
n=5: [1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1];
n=6: [1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 1]; ...
This sequence gives the sums of the absolute values of the coefficients for n>=0.

Cf. A192210, A084611, A084613, A084615, A084775, A084776, A084777, A084778, A084779, A084780.

Cf. A227964 (triangle).

Table[Total[Abs[CoefficientList[Expand[(1-x-x^2+x^3)^n],x]]],{n,0,30}] (* Harvey P. Dale, Mar 03 2013 *)

{a(n)=sum(k=0,3*n,abs(polcoeff((1-x-x^2+x^3)^n,k)))}
for(n=0,30,print1(a(n),", "))

A192210 a(n) = sum of unsigned coefficients in (1+x+x^2-x^3)^n.

Original entry on oeis.org

1, 4, 10, 26, 80, 194, 504, 1442, 3710, 9536, 26842, 69014, 178704, 496602, 1316204, 3377206, 9242898, 24629944, 63304540, 172497622, 462822414, 1210912388, 3177522724, 8736822276, 22617998204, 59776061150, 163702751968, 433787373560

Offset: 0

Paul D. Hanna, Jun 25 2011

nonn

What is the behavior of this sequence? Does there exist a g.f.?
It would be nice to know the (more accurate) values of the following limits:
(1) The position of the first negative coefficient in (1+x+x^2-x^3)^n, divided by n, seems to reach a limit near 0.398...
(2) Limit a(n)^(1/n) seems to exist near 2.6637...
(3) Limit a(n+1)/a(n) does not seem to be unique; attractors seem to exist near 2.66...

			Illustrate the coefficients in (1+x+x^2-x^3)^n by:
n=0: [1];
n=1: [1, 1, 1, -1];
n=2: [1, 2, 3, 0, -1, -2, 1];
n=3: [1, 3, 6, 4, 0, -6, -2, 0, 3, -1];
n=4: [1, 4, 10, 12, 7, -8, -12, -8, 7, 4, 2, -4, 1];
n=5: [1, 5, 15, 25, 25, 1, -25, -35, -5, 15, 21, -5, -5, -5, 5, -1];
n=6: [1, 6, 21, 44, 60, 36, -24, -84, -66, 0, 66, 36, -4, -36, 0, 4, 9, -6, 1];
n=7: [1, 7, 28, 70, 119, 119, 28, -132, -210, -126, 84, 168, 98, -70, -76, -28, 49, 7, 0, -14, 7, -1]; ...
This sequence gives the sums of the absolute values of the coefficients for n>=0.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A192205, A084611, A084613, A084615, A084775, A084776, A084777, A084778, A084779, A084780.

Table[Total[Abs[CoefficientList[Expand[(1+x+x^2-x^3)^n],x]]],{n,0,30}] (* Harvey P. Dale, Oct 12 2012 *)

{a(n)=sum(k=0,3*n,abs(polcoeff((1+x+x^2-x^3)^n,k)))}

cp's OEIS Frontend

A084776 a(n) = sum of absolute-valued coefficients of (1+2*x-x^2)^n.

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A084777 a(n) = sum of absolute-valued coefficients of (1+2*x-2*x^2)^n.

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A084778 a(n) = sum of absolute-valued coefficients of (1+2*x-3*x^2)^n.

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A084779 a(n) = sum of absolute-valued coefficients of (1+2*x-4*x^2)^n.

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A084780 a(n) = sum of absolute-valued coefficients of (1+3*x-x^2)^n.

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A192205 a(n) = sum of absolute values of coefficients in (1-x-x^2+x^3)^n.

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A192210 a(n) = sum of unsigned coefficients in (1+x+x^2-x^3)^n.

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A084777 a(n) = sum of absolute-valued coefficients of (1+2x-2x^2)^n.

A084778 a(n) = sum of absolute-valued coefficients of (1+2x-3x^2)^n.

A084779 a(n) = sum of absolute-valued coefficients of (1+2x-4x^2)^n.