A084600 -id:A084600 - cp's OEIS frontend

A084613 a(n) = sum of absolute values of coefficients of (1 + x - 2*x^2)^n.

1, 4, 14, 44, 124, 394, 1418, 4706, 14322, 40712, 135878, 468934, 1513650, 4502864, 13421408, 45258442, 152708520, 483810550, 1413811358, 4483843328, 15051967962, 49724234652, 154802614364, 461020649750, 1486736569982

Offset: 0

Paul D. Hanna, Jun 01 2003

nonn

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

Cf. A002426, A084600 to A084612, A084614, A084615.

A084612:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-2)^j: j in [0..k]]) >;
[(&+[Abs(A084612(n,k)): k in [0..2*n]]): n in [0..50]]; // G. C. Greubel, Mar 25 2023

Table[Total[Abs[CoefficientList[Expand[(1+x-2x^2)^n],x]]],{n,0,30}]  (* Harvey P. Dale, Apr 21 2011 *)

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

def A084612(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-2)^j for j in range(k+1))
def A084613(n): return sum(abs(A084612(n,k)) for k in range(2*n+1))
[A084613(n) for n in range(51)] # G. C. Greubel, Mar 25 2023

A084614 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 3x^2)^n.

Original entry on oeis.org

1, 1, 1, -3, 1, 2, -5, -6, 9, 1, 3, -6, -17, 18, 27, -27, 1, 4, -6, -32, 19, 96, -54, -108, 81, 1, 5, -5, -50, 5, 211, -15, -450, 135, 405, -243, 1, 6, -3, -70, -30, 366, 181, -1098, -270, 1890, -243, -1458, 729, 1, 7, 0, -91, -91, 546, 637, -2015, -1911, 4914, 2457, -7371, 0, 5103, -2187, 1, 8, 4, -112, -182, 728, 1456

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
  1;
  1, 1, -3;
  1, 2, -5,  -6,   9;
  1, 3, -6, -17,  18,  27, -27;
  1, 4, -6, -32,  19,  96, -54,  -108,   81;
  1, 5, -5, -50,   5, 211, -15,  -450,  135,  405, -243;
  1, 6, -3, -70, -30, 366, 181, -1098, -270, 1890, -243, -1458, 729;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A002426, A027471, A084600 - A084613, A084615.

A084614:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-3)^j: j in [0..k]]) >;
[A084614(n,k): k in [0..2*n], n in [0..15]]; // G. C. Greubel, Mar 25 2023

With[{eq= (1+x-3*x^2)}, Flatten[Table[CoefficientList[Expand[eq^n], x], {n,0,13}]]] (* G. C. Greubel, Mar 02 2017 *)

for(n=0,12, for(k=0,2*n,t=polcoeff((1+x-3*x^2)^n,k,x); print1(t",")); print(" "))

def A084614(n,k): return ( (1+x-3*x^2)^n ).series(x, 30).list()[k]
flatten([[A084614(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Mar 25 2023

From G. C. Greubel, Mar 25 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*(-3)^j, for 0 <= k <= 2*n.
T(n, 2*n) = (-3)^n.
T(n, 2*n-1) = (-1)^(n-1)*A027471(n+1), n >= 1.
Sum_{k=0..2*n} T(n, k) = (-1)^n.
Sum_{k=0..2*n} (-1)^k*T(n, k) = (-3)^n. (End)

A084602 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 3x^2)^n.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 7, 6, 9, 1, 3, 12, 19, 36, 27, 27, 1, 4, 18, 40, 91, 120, 162, 108, 81, 1, 5, 25, 70, 185, 331, 555, 630, 675, 405, 243, 1, 6, 33, 110, 330, 726, 1441, 2178, 2970, 2970, 2673, 1458, 729, 1, 7, 42, 161, 539, 1386, 3157, 5797, 9471, 12474, 14553

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
{1},
{1,1, 3},
{1,2, 7,  6,  9},
{1,3,12, 19, 36, 27,  27},
{1,4,18, 40, 91,120, 162, 108,  81},
{1,5,25, 70,185,331, 555, 630, 675, 405, 243},
{1,6,33,110,330,726,1441,2178,2970,2970,2673,1458,729},

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A002426, A084600-A084601, A084603-A084615.

With[{eq = (1 + x + 3*x^2)}, Flatten[Table[CoefficientList[Expand[eq^n], x], {n, 0, 10}]]] (* G. C. Greubel, Mar 02 2017 *)

for(n=0,15, for(k=0,2*n,t=polcoeff((1+x+3*x^2)^n,k,x); print1(t",")); print(" "))

A084604 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 4x^2)^n.

Original entry on oeis.org

1, 1, 1, 4, 1, 2, 9, 8, 16, 1, 3, 15, 25, 60, 48, 64, 1, 4, 22, 52, 145, 208, 352, 256, 256, 1, 5, 30, 90, 285, 561, 1140, 1440, 1920, 1280, 1024, 1, 6, 39, 140, 495, 1206, 2841, 4824, 7920, 8960, 9984, 6144, 4096, 1, 7, 49, 203, 791, 2261, 6027, 12489, 24108, 36176

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
{1},
{1,1, 4},
{1,2, 9,  8, 16},
{1,3,15, 25, 60,  48,  64},
{1,4,22, 52,145, 208, 352, 256, 256},
{1,5,30, 90,285, 561,1140,1440,1920,1280,1024},
{1,6,39,140,495,1206,2841,4824,7920,8960,9984,6144,4096},

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A002426, A084600-A084603, A084605-A084615.

With[{eq=(1+x+4x^2)},Flatten[Table[CoefficientList[Expand[eq^n],x],{n,0,10}]]] (* Harvey P. Dale, May 19 2011 *)

for(n=0,10, for(k=0,2*n,t=polcoeff((1+x+4*x^2)^n,k,x); print1(t",")); print(" "))

A084612 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 2*x^2)^n.

Original entry on oeis.org

1, 1, 1, -2, 1, 2, -3, -4, 4, 1, 3, -3, -11, 6, 12, -8, 1, 4, -2, -20, 1, 40, -8, -32, 16, 1, 5, 0, -30, -15, 81, 30, -120, 0, 80, -32, 1, 6, 3, -40, -45, 126, 141, -252, -180, 320, 48, -192, 64, 1, 7, 7, -49, -91, 161, 357, -363, -714, 644, 728, -784, -224, 448, -128, 1, 8, 12, -56, -154, 168, 700, -328, -1791, 656, 2800

Offset: 0

Paul D. Hanna, Jun 01 2003

			Triangle begins:
  1;
  1, 1, -2;
  1, 2, -3,  -4,   4;
  1, 3, -3, -11,   6,  12,  -8;
  1, 4, -2, -20,   1,  40,  -8,  -32,   16;
  1, 5,  0, -30, -15,  81,  30, -120,    0,  80, -32;
  1, 6,  3, -40, -45, 126, 141, -252, -180, 320,  48, -192, 64;

Paul D. Hanna, Table of n, a(n) for n = 0..1023

Cf. A000007, A001787, A002426, A084600 - A084611, A084613, A084614, A084615.

A084612:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-2)^j: j in [0..k]]) >;
[A084612(n,k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 25 2023

T[n_, k_]:= Sum[Binomial[n,k-j]*Binomial[k-j,j]*(-2)^j, {j,0,k}];
Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Mar 25 2023 *)

{T(n,k)=polcoeff((1+x-2*x^2)^n, k)}
for(n=0,10,for(k=0,2*n,print1(T(n,k),", "));print(""))

def A084612(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-2)^j for j in range(k+1))
flatten([[A084612(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Mar 25 2023

From G. C. Greubel, Mar 25 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n,k-j)*binomial(k-j,j)*(-2)^j, for 0 <= k <= 2*n.
T(n, 2*n) = (-2)^n.
T(n, 2*n-1) = (-1)^(n-1)*A001787(n), n >= 1.
Sum_{k=0..2*n} T(n, k) = A000007(n).
Sum_{k=0..2*n} (-1)^k*T(n, k) = (-2)^n. (End)

A251687 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} T(n,k)^2 x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + x + 2*x^2)^n.

Original entry on oeis.org

1, 1, 1, 5, 8, 8, 16, 28, 48, 80, 128, 208, 320, 512, 768, 1216, 1792, 2816, 4096, 6400, 9216, 14336, 20480, 31744, 45056, 69632, 98304, 151552, 212992, 327680, 458752, 704512, 983040, 1507328, 2097152, 3211264, 4456448, 6815744, 9437184, 14417920, 19922944, 30408704, 41943040

Offset: 0

Paul D. Hanna, Mar 03 2015

nonn

More generally, if G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (p + q*x + r*x^2)^n, then G(x) = (1 + p^2*x)*(1 + r^2*x^3)*(1 + (q^2-2*p*r)*x^2 + p^2*r^2*x^4) / (1-p*r*x^2)^2.

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 8*x^4 + 8*x^5 + 16*x^6 + 28*x^7 +...
where
log(A(x)) = (1 + x + 2^2*x^2)/A(x) * x +
(1 + 2^2*x + 5^2*x^2 + 4^2*x^3 + 4^2*x^4)/A(x)^2 * x^2/2 +
(1 + 3^2*x + 9^2*x^2 + 13^2*x^3 + 18^2*x^4 + 12^2*x^5 + 8^2*x^6)/A(x)^3 * x^3/3 +
(1 + 4^2*x + 14^2*x^2 + 28^2*x^3 + 49^2*x^4 + 56^2*x^5 + 56^2*x^6 + 32^2*x^7 + 16^2*x^8)/A(x)^4 * x^4/4 +
(1 + 5^2*x + 20^2*x^2 + 50^2*x^3 + 105^2*x^4 + 161^2*x^5 + 210^2*x^6 + 200^2*x^7 + 160^2*x^8 + 80^2*x^9 + 32^2*x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients in (1 + x + 2*x^2)^n - see triangle A084600.

Cf. A084600, A251688, A251689, A200537.

/* By Definition: */
{a(n,p=1,q=1,r=2)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, polcoeff((p + q*x + r*x^2 +x*O(x^k))^m, k)^2 *x^k) *x^m/(A+x*O(x^n))^m/m)+x*O(x^n))); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

/* By G.F. Identity (faster): */
{a(n,p=1,q=1,r=2)=polcoeff( (1 + p^2*x)*(1 + r^2*x^3)*(1 + (q^2-2*p*r)*x^2 + p^2*r^2*x^4) / ((1-p*r*x^2)^2 +x*O(x^n)), n)}
for(n=0, 50, print1(a(n), ", "))

G.f.: (1 + x)*(1 + 4*x^3)*(1 - 3*x^2 + 4*x^4) / (1 - 2*x^2)^2.

cp's OEIS Frontend

A084613 a(n) = sum of absolute values of coefficients of (1 + x - 2*x^2)^n.

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A084614 Triangle, read by rows, where the n-th row lists the (2*n+1) coefficients of (1 + x - 3*x^2)^n.

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A084602 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 3x^2)^n.

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A084604 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 4x^2)^n.

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A084612 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 2*x^2)^n.

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A251687 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + x + 2*x^2)^n.

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A084614 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 3x^2)^n.

A251687 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} T(n,k)^2 x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + x + 2*x^2)^n.