A027907 -id:A027907 - cp's OEIS frontend

A186236 G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2n} A027907(n,k)^2 x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

1, 1, 2, 5, 13, 34, 93, 262, 753, 2198, 6502, 19449, 58724, 178739, 547836, 1689407, 5237939, 16318137, 51056027, 160363129, 505456920, 1598263936, 5068483189, 16116397411, 51371962474, 164123564499, 525447953073, 1685534207788, 5416719384326, 17437073203711

Offset: 0

Paul D. Hanna, Oct 19 2011

nonn

Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 34*x^5 + 93*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 91*x^5/5 + 282*x^6/6 + 890*x^7/7 + 2831*x^8/8 + 9055*x^9/9 + 29133*x^10/10 +...
which equals the sum of the series:
log(A(x)) = (1 + x + x^2)*x
+ (1 + 2^2*x + 3^2*x^2 + 2^2*x^3 + x^4)*x^2/2
+ (1 + 3^2*x + 6^2*x^2 + 7^2*x^3 + 6^2*x^4 + 3*x^5 + x^6)*x^3/3
+ (1 + 4^2*x + 10^2*x^2 + 16^2*x^3 + 19^2*x^4 + 16^2*x^5 + 10^2*x^6 + 4^2*x^7 + x^8)*x^4/4
+ (1 + 5^2*x + 15^2*x^2 + 30^2*x^3 + 45^2*x^4 + 51^2*x^5 + 45^2*x^6 + 30^2*x^7 + 15^2*x^8 + 5^2*x^9 + x^10)*x^5/5 +...

Cf. A180718 (variant).

{A027907(n,k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, 2*m, A027907(m,k)^2 *x^k) *x^m/m)+x*O(x^n)), n)}

A099602 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907), omitting leading zeros.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 4, 1, 1, 5, 8, 5, 1, 3, 13, 22, 18, 7, 1, 1, 9, 26, 35, 24, 8, 1, 4, 26, 70, 101, 84, 40, 10, 1, 1, 14, 61, 131, 160, 116, 49, 11, 1, 5, 45, 171, 363, 476, 400, 215, 71, 13, 1, 1, 20, 120, 363, 654, 752, 565, 275, 83, 14, 1, 6, 71, 356, 1017, 1856, 2282, 1932

Offset: 0

Paul D. Hanna, Oct 25 2004

Row sums form A099603, where A099603(n) = Fibonacci(n+1)*2^floor((n+1)/2). Central coefficients of even-indexed rows form A082759, where A082759(n) = Sum_{k=0..n} binomial(n,k)*trinomial(n,k). Antidiagonal sums form A099604.

Matrix inverse equals triangle A104495, which is generated from self-convolutions of the Catalan sequence (A000108).

			Rows begin:
  1;
  1,  1;
  1,  2,   1;
  2,  5,   4,   1;
  1,  5,   8,   5,   1;
  3, 13,  22,  18,   7,   1;
  1,  9,  26,  35,  24,   8,   1;
  4, 26,  70, 101,  84,  40,  10,   1;
  1, 14,  61, 131, 160, 116,  49,  11,  1;
  5, 45, 171, 363, 476, 400, 215,  71, 13,  1;
  1, 20, 120, 363, 654, 752, 565, 275, 83, 14, 1;
  ...
The binomial transform of row 2 = column 2 of A027907: BINOMIAL[1,2,1] = [1,3,6,10,15,21,28,36,45,55,...].
The binomial transform of row 3 = column 3 of A027907: BINOMIAL[2,5,4,1] = [2,7,16,30,50,77,112,156,210,...].
The binomial transform of row 4 = column 4 of A027907: BINOMIAL[1,5,8,5,1] = [1,6,19,45,90,161,266,414,615,...].
The binomial transform of row 5 = column 5 of A027907: BINOMIAL[3,13,22,18,7,1] = [3,16,51,126,266,504,882,1452,...].

Cf. A027907, A082759, A099603, A099604.

Cf. A104495.

{T(n,k)=polcoeff(polcoeff((1+(y+1)*x-(y+1)*x^2)/(1-(y+1)*(y+2)*x^2+(y+1)^2*x^4)+x*O(x^n),n,x)+y*O(y^k),k,y)}

{T(n,k)=(matrix(n+1,n+1,i,j,if(i>=j,polcoeff(polcoeff( (1+x*y/(1+x))/(1+x-y^2*(1-(1+4*x+O(x^i))^(1/2))^2/4+O(y^j)),i-1,x),j-1,y)))^-1)[n+1,k+1]}

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

A168590 G.f.: exp( Sum_{n>=1} A168591(n)*x^n/n ), where A168591(n) = sum of the n-th power of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 14, 310, 71399, 153056789, 2826352872319, 445742192193898313, 602479884829000885595175, 7000510736697461064666950774905, 701725717683874683612335083605682943282

Offset: 0

Paul D. Hanna, Dec 01 2009

nonn

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 310*x^3 + 71399*x^4 +...
log(A(x)) = 3*x + 19*x^2/2 + 831*x^3/3 + 281907*x^4/4 +...+ A168591(n)*x^n/n +...

Cf. A168591, A168592, A168593, A027907.

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sum(k=0,2*m,polcoeff((1+x+x^2)^m,k)^m)*x^m/m) +x*O(x^n)),n))}

A168595 a(n) = Sum_{k=0..2n} C(2n,k)*A027907(n,k) where A027907 is the triangle of trinomial coefficients.

Original entry on oeis.org

1, 4, 36, 358, 3748, 40404, 443886, 4941654, 55555236, 629285416, 7170731236, 82108083204, 943960439086, 10889085499348, 125974782200478, 1461030555025458, 16981658850393252, 197757344280343968

Offset: 0

Paul D. Hanna, Nov 30 2009

nonn

Compare to A092765(n) = Sum_{k=0..2n} (-1)^k*C(2n,k)*A027907(n,k), which is the number of paths of length n ending at origin in 1-D random walk with jumps to next-nearest neighbors.

Cf. A027907, A092765.

cb := n -> binomial(2*n, n);
a := n -> add((-1)^(n-k)*binomial(n,k)*cb(n+k), k=0..n);
seq(a(n), n=0..17); # Peter Luschny, Aug 15 2017

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

a(n) = 2*A132306(n) for n > 0. - Mark van Hoeij, Jul 02 2010

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*cb(n+k) with cb(n) = binomial(2n,n). - Peter Luschny, Aug 15 2017

A027913 T(n,[ n/2 ]), T given by A027907.

Original entry on oeis.org

1, 1, 2, 3, 10, 15, 50, 77, 266, 414, 1452, 2277, 8074, 12727, 45474, 71955, 258570, 410346, 1481108, 2355962, 8533660, 13599915, 49402850, 78855339, 287134346, 458917850, 1674425300, 2679183405, 9792273690, 15683407785

Offset: 0

Clark Kimberling

nonn

The median coefficient in the expansion of (1 + x + x^2)^n. - Vladimir Reshetnikov, Nov 21 2020

Robert Israel, Table of n, a(n) for n = 0..2550

Cf. A027907, A027908.

seq(simplify(GegenbauerC(floor(n/2),-n,-1/2)), n=0..100); # Robert Israel, Oct 20 2016

Table[GegenbauerC[Floor[n/2], -n, -1/2] + KroneckerDelta[n, 0], {n, 0,
100}] (* Emanuele Munarini, Oct 20 2016 *)

makelist(ultraspherical(floor(n/2),-n,-1/2),n,0,12); /* Emanuele Munarini, Oct 18 2016 */

a(n) = GegenbauerC(floor(n/2), -n, -1/2). - Emanuele Munarini, Oct 18 2016

G.f.: g(t) = (1+(t+t^2)*A(t^2)+t^4*A(t^2)^2)/(1-t^2*A(t^2)-3*t^4*A(t^2)^2), where A(t) is the g.f. of A143927 and satisfies A(t) = [1 + x*A(t) + t^2*A(t)^2]^2. - Emanuele Munarini, Oct 20 2016

A099603 Row sums of triangle A099602, in which row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

Original entry on oeis.org

1, 2, 4, 12, 20, 64, 104, 336, 544, 1760, 2848, 9216, 14912, 48256, 78080, 252672, 408832, 1323008, 2140672, 6927360, 11208704, 36272128, 58689536, 189923328, 307302400, 994451456, 1609056256, 5207015424, 8425127936, 27264286720, 44114542592, 142757658624, 230986743808

Offset: 0

Paul D. Hanna, Oct 25 2004

			Sequence begins: {1*1, 1*2, 2*2, 3*4, 5*4, 8*8, 13*8, 21*16, 34*16, ...}.

Stefano Spezia, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-4).

Cf. A000045, A027907, A099602.

LinearRecurrence[{0,6,0,-4},{1,2,4,12},30] (* Harvey P. Dale, Aug 09 2016 *)

PARI
```
a(n)=fibonacci(n+1)*2^((n+1)\2)
```

a(n) = Fibonacci(n+1)*2^((n+1)/2).
a(n) = 6*a(n-2) - 4*a(n-4) for n>4.
G.f.: (1+2*x-2*x^2)/(1-6*x^2+4*x^4).

A168593 G.f.: exp( Sum_{n>=1} A132303(n)*x^n/n ), where A132303(n) = sum of the cubes of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 27, 349, 5484, 96408, 1824758, 36393090, 754696998, 16130052394, 353134333470, 7884110379006, 178908263232959, 4115917059924057, 95806493175049929, 2252809457441037107, 53443567449376649304

Offset: 0

Paul D. Hanna, Dec 01 2009

nonn

Self-convolution cube-root yields the integer sequence A251686.

			G.f.: A(x) = 1 + 3*x + 27*x^2 + 349*x^3 + 5484*x^4 + 96408*x^5 +...
log(A(x)) = 3*x + 45*x^2/2 + 831*x^3/3 + 17181*x^4/4 + 375903*x^5/5 +...+ A132303(n)*x^n/n +...

Cf. A168590, A168592, A132303, A027907, A251686.

{A027907(n,k) = polcoeff((1+x+x^2)^n, k)}
{A132303(n) = sum(k=0, 2*n, A027907(n,k)^3)}
{a(n) = local(A); A = exp(sum(m=1, n+1, A132303(m)*x^m/m) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A199248 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} A027907(n,k)^2 x^k * A(x)^k]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

Original entry on oeis.org

1, 1, 2, 6, 20, 69, 248, 923, 3523, 13706, 54152, 216710, 876607, 3578405, 14722432, 60986158, 254145337, 1064712328, 4481577078, 18943753140, 80381689202, 342254333393, 1461864544896, 6262021627055, 26894816382199, 115792035533779, 499648608539714, 2160504474956390

Offset: 0

Paul D. Hanna, Nov 04 2011

nonn

Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 69*x^5 + 248*x^6 + 923*x^7 +...
such that A(x) = G(x*A(x)) where G(x) is given by:
G(x) = (1 - x + x^2)*(1 - x^2 + x^4)/(1-x)^2 = (1-x^5)/(1-x) + x^3/(1-x)^2:
G(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 7*x^9 +...
...
Let A = x*A(x), then the logarithm of the g.f. A(x) equals the series:
log(A(x)) = (1 + A + A^2)*x +
(1 + 2^2*A + 3^2*A^2 + 2^2*A^3 + A^4)*x^2/2 +
(1 + 3^2*A + 6^2*A^2 + 7^2*A^3 + 6^2*A^4 + 3^2*A^5 + A^6)*x^3/3 +
(1 + 4^2*A + 10^2*A^2 + 16^2*A^3 + 19^2*A^4 + 16^2*A^5 + 10^2*A^6 + 4^2*A^7 + A^8)*x^4/4 +
(1 + 5^2*A + 15^2*A^2 + 30^2*A^3 + 45^2*A^4 + 51^2*A^5 + 45^2*A^6 + 30^2*A^7 + 15^2*A^8 + 5^2*A^9 + A^10)*x^5/5 +...
which involves the squares of the trinomial coefficients A027907(n,k).

Cf. A186236, A199257, A027907.

{a(n)=local(A=1+x); A=1/x*serreverse(x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12+x*O(x^n))); polcoeff(A, n)}

/* G.f. A(x) using the squares of the trinomial coefficients */
{A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k*A^k) *x^m/m)+x*O(x^n)));polcoeff(A, n)}

G.f. satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1 - x + x^2)*(1 - x^2 + x^4)/(1-x)^2.

G.f.: A(x) = (1/x)*Series_Reversion( x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12) ).

A027915 a(n) = Sum_{0<=j<=i, 0<=i<=n} A027907(i, j).

Original entry on oeis.org

1, 3, 9, 26, 76, 223, 658, 1948, 5782, 17193, 51194, 152594, 455209, 1358841, 4058439, 12126696, 36248370, 108385917, 324172566, 969801726, 2901883611, 8684735577, 25995833145, 77824036620, 233012973051, 697745695923

Offset: 0

Clark Kimberling

nonn

Partial sums of A027914.

G.f.: (1+x+1/G(0))/(2*(1-2*x-3*x^2))/(1-x), where G(k)= 1 + x*(2+3*x)*(4*k+1)/(4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 30 2013

A099604 Antidiagonal sums of triangle A099602, in which row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 23, 40, 72, 131, 233, 420, 756, 1355, 2438, 4381, 7868, 14144, 25413, 45661, 82058, 147444, 264943, 476092, 855483, 1537236, 2762296, 4963591, 8919173, 16027012, 28799164, 51749715, 92989886, 167094985, 300255720

Offset: 0

Paul D. Hanna, Oct 25 2004

nonn

Index entries for linear recurrences with constant coefficients, signature (0, 2, 3, 0, -2, -1).

Cf. A099602, A027907.

LinearRecurrence[{0, 2, 3, 0, -2, -1}, {1, 1, 2, 4, 7, 12}, 35] (* Jean-François Alcover, Oct 30 2017 *)

a(n)=polcoeff((1+x-x^3)/(1-2*x^2-3*x^3+2*x^5+x^6)+x*O(x^n),n,x)

G.f.: (1+x-x^3)/(1-2*x^2-3*x^3+2*x^5+x^6).

a(n) = 2*a(n-2) + 3*a(n-3) - 2*a(n-5) - a(n-6) for n>=6.

cp's OEIS Frontend

A186236 G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2*n} A027907(n,k)^2 * x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

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A099602 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907), omitting leading zeros.

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A168590 G.f.: exp( Sum_{n>=1} A168591(n)*x^n/n ), where A168591(n) = sum of the n-th power of the trinomial coefficients in row n of triangle A027907.

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A168595 a(n) = Sum_{k=0..2n} C(2n,k)*A027907(n,k) where A027907 is the triangle of trinomial coefficients.

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Maple

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A027913 T(n,[ n/2 ]), T given by A027907.

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Mathematica

Maxima

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A099603 Row sums of triangle A099602, in which row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

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A168593 G.f.: exp( Sum_{n>=1} A132303(n)*x^n/n ), where A132303(n) = sum of the cubes of the trinomial coefficients in row n of triangle A027907.

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A199248 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A027907(n,k)^2 * x^k * A(x)^k]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

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A186236 G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2n} A027907(n,k)^2 x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

A199248 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2n} A027907(n,k)^2 x^k * A(x)^k]* x^n/n ), where A027907 is the triangle of trinomial coefficients.