A027907 -id:A027907 - cp's OEIS frontend

A251686 G.f.: exp( Sum_{n>=1} A132303(n)/3 * x^n/n ), where A132303(n) = sum of the cubes of the trinomial coefficients in row n of triangle A027907.

1, 1, 8, 100, 1556, 27260, 515510, 10284094, 213433728, 4566363088, 100082133066, 2236952393302, 50817223209451, 1170319824912699, 27268900054818390, 641812268110993694, 15239341125950643462, 364655982858022960206, 8785745372509009963892, 212976842702489760621536

Offset: 0

Paul D. Hanna, Feb 28 2015

nonn

Self-convolution cube yields A168593.

			G.f.: A(x) = 1 + x + 8*x^2 + 100*x^3 + 1556*x^4 + 27260*x^5 +...
where
log(A(x)) = 1*x + 15*x^2/2 + 277*x^3/3 + 5727*x^4/4 + 125301*x^5/5 + 2843643*x^6/6 + 66214485*x^7/7 + 1571497119*x^8/8 +...+ A132303(n)/3*x^n/n +...

Cf. A168593, A132303, A027907.

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

A027909 T(2n,n-1), T given by A027907.

Original entry on oeis.org

1, 4, 21, 112, 615, 3432, 19383, 110448, 633726, 3656360, 21191555, 123286440, 719539015, 4210967880, 24702429825, 145210795200, 855172338570, 5044470461352, 29799593861974, 176268499363840, 1043889366927771, 6188748520285584, 36726461258947569, 218146172715259488, 1296812014083995850

Offset: 0

Clark Kimberling

nonn

seq(add(binomial(j,2*j-3*n-4)*binomial(2*n+2,j),j=0...2*n+2),n=0..25);  # Mark van Hoeij, May 12 2013

a(n)=sum(j=0,2*n+2,binomial(j, 2*j-3*n-4)*binomial(2*n+2, j)); \\ Joerg Arndt, May 13 2013~

G.f.: -g*(g^2+g+1)/(3*g^2+g-1) where g = x times the g.f. of A143927. - Mark van Hoeij, Nov 16 2011

Corrected offset, more terms, Joerg Arndt, May 13 2013

A027911 a(n) = T(2*n+1,n), with T given by A027907.

Original entry on oeis.org

1, 3, 15, 77, 414, 2277, 12727, 71955, 410346, 2355962, 13599915, 78855339, 458917850, 2679183405, 15683407785, 92022516525, 541050073146, 3186886397310, 18801598011274, 111083331666918, 657153430251396, 3892199032434105, 23077435617920925, 136963282273730613, 813597690808666386

Offset: 0

Clark Kimberling

nonn

Cf. A027907, A143927.

seq(add(binomial(j,2*j-2-3*n)*binomial(2*n+1,j),j=0...2*n+1),n=0..20);  # Mark van Hoeij, May 12 2013

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

makelist(ultraspherical(n,-2*n-1,-1/2),n,0,12); /* Emanuele Munarini, Oct 20 2016 */

a(n)=sum(j=0, 2*n+1, binomial(j, 2*j-2-3*n)*binomial(2*n+1, j)); \\ Joerg Arndt, Oct 20 2016

a(n) = GegenbauerPoly(n,-2*n-1,-1/2). - Emanuele Munarini, Oct 20 2016
G.f.: g/(1-g-3*g^2), where g = x times the g.f. of A143927. - Mark van Hoeij, Nov 16 2011
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+1,k)*binomial(2*n+1-k,n-2*k). - Emanuele Munarini, Oct 20 2016

More terms from Joerg Arndt, Oct 20 2016

A027912 T(2n-1,n-2), T given by A027907.

Original entry on oeis.org

1, 5, 28, 156, 880, 5005, 28665, 165104, 955434, 5550755, 32355917, 189147400, 1108476720, 6510243495, 38308997100, 225810489168, 1333057076890

Offset: 2

Clark Kimberling

nonn

A059781 Triangle T(n,k) giving exponent of power of 2 dividing entry (n,k) of trinomial triangle A027907.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 4, 0, 4, 1, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 2, 4, 1, 3, 4, 3, 0, 3, 4, 3, 1, 4, 2, 3, 0, 0, 0, 0, 2, 1, 1, 1, 8, 0, 0, 0, 8, 1, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane, Feb 22 2001

			0; 0,0,0; 0,1,0,1,0; ...

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 117.

with(numtheory): T := proc(i,j) option remember: if i >= 0 and j=0 then RETURN(1) fi: if i >= 0 and j=2*i then RETURN(1) fi: if i >= 1 and j=1 then RETURN(i) fi: if i >= 1 and j=2*i-1 then RETURN(i) fi: T(i-1,j-2)+T(i-1,j-1)+T(i-1,j): end: for i from 0 to 20 do for j from 0 to 2*i do if T(i,j) mod 2 = 1 then printf(`%d,`,0) else printf(`%d,`, ifactors(T(i,j))[2,1,2] ) fi: od:od: # James Sellers, Feb 22 2001

More terms from James Sellers, Feb 22 2001

A059782 Triangle T(n,k) giving exponent of power of 3 dividing entry (n,k) of trinomial triangle A027907.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 0, 0, 1, 1, 0, 2, 2, 1, 2, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 2, 1, 2, 2, 1, 2, 2, 0, 2, 2, 1, 2, 2, 1, 2, 2, 0, 0, 0, 0, 1, 1

Offset: 0

N. J. A. Sloane, Feb 22 2001

			0; 0,0,0; 0,0,1,0,0; 0,1,1,0,1,1,0; ...

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 118.

with(numtheory): T := proc(i,j) option remember: if i >= 0 and j=0 then RETURN(1) fi: if i >= 0 and j=2*i then RETURN(1) fi: if i >= 1 and j=1 then RETURN(i) fi: if i >= 1 and j=2*i-1 then RETURN(i) fi: T(i-1,j-2)+T(i-1,j-1)+T(i-1,j): end: for i from 0 to 20 do for j from 0 to 2*i do if T(i,j) mod 3 <> 0 then printf(`%d,`,0) fi: if T(i,j) mod 3 = 0 and T(i,j) mod 2 = 0 then printf(`%d,`, ifactors(T(i,j))[2,2,2] ) fi: if T(i,j) mod 3 = 0 and T(i,j) mod 2 = 1 then printf(`%d,`, ifactors(T(i,j))[2,1,2] ) fi: #printf(`%d,`,T(i,j)) od:od: # James Sellers, Feb 22 2001

More terms from James Sellers, Feb 22 2001

A101617 The trinomial transform (A027907) gives powers of 3, while the trinomial transform of this sequence shift one place left gives powers of 5.

Original entry on oeis.org

1, 1, 1, 3, -3, 19, -43, 139, -355, 995, -2587, 6907, -17939, 46931, -121419, 314603, -811203, 2091459, -5379963, 13833179, -35527795, 91210035, -234020267, 600258507, -1539135779, 3945762211, -10113490139, 25918908603, -66417608403, 170182721299, -436032111883, 1117120911019

Offset: 0

Paul D. Hanna, Dec 09 2004

sign

			3^3 = 1*(1) + 3*(1) + 6*(1) + 7*(3) + 6*(-3) + 3*(19) + 1*(-43).
5^3 = 1*(1) + 3*(1) + 6*(3) + 7*(-3) + 6*(19) + 3*(-43) + 1*(139).
In general, a sequence A with the property that the
trinomial transform of A gives powers of P, while the
trinomial transform of LSHIFT(A) gives powers of Q
has the g.f.: N(x)/D(x) where
N(x)=(1+3*x-(Q-3)*x^2-(P+Q-2)*x^3) and
D(x)=(1+2*x-(P+Q-3)*x^2-(P+Q-2)*x^3+(P-1)*(Q-1)*x^4).

Cf. A027907, A100321.

nn:=31; CoefficientList[Series[(1 + 3*x - 2*x^2 - 6*x^3)/(1 + 2*x - 5*x
^2 - 6*x^3 + 8*x^4),{x,0,nn}],x] (* Georg Fischer, Apr 17 2020 *)

{a(n)=local(P=3,Q=5,V=[1,1]);if(n>1, for(m=1,n, V=concat(V,P^m-sum(k=0,2*m-1,polcoeff((1+x+x^2)^m+x*O(x^k),k)*V[k+1])); V=concat(V,Q^m-sum(k=0,2*m-1,polcoeff((1+x+x^2)^m+x*O(x^k),k)*V[k+2])); ));V[n+1]}
for(n=0,30,print1(a(n),", "))

G.f.: A(x) = (1 + 3*x - 2*x^2 - 6*x^3)/(1 + 2*x - 5*x^2 - 6*x^3 + 8*x^4). [corrected by Georg Fischer, Apr 17 2020]
3^n = Sum_{k=0..2*n} A027907(n, k)*a(k) for n>=0 and
5^n = Sum_{k=0..2*n} A027907(n, k)*a(k+1) for n>=0.
a(n) = (-1)^n*A006131(n-1) + (1/3)[(-2)^n + 2]. - Ralf Stephan, May 16 2007

A123934 Triangle T(n,k), 1<=k<=n :forms the odd-indexed trinomial coefficients (A027907).

Original entry on oeis.org

1, 2, 2, 3, 7, 3, 4, 16, 16, 4, 5, 30, 51, 30, 5, 6, 50, 126, 126, 50, 6, 7, 77, 266, 393, 266, 77, 7

Offset: 1

Philippe Deléham, Oct 30 2006

			Triangle begins:
1;
2, 2;
3, 7, 3;
4, 16, 16, 4;
5, 30, 51, 30, 5;
6, 50, 126, 126, 50, 6;
7, 77, 266, 393, 266, 77, 7;

Cf. A056241.

Sum_{k, 1<=k<=n}T(n,k)=A003462(n)=(3^n-1)/2.

A136168 a(n) = (n-1)!Sum_{i=1..n-1} (-1)^(i+1)A027907(n-i+2,i+1)*a(n-i)/(n-i)! for n>0 with a(0)=1, where A027907 is the triangle of trinomial coefficients.

Original entry on oeis.org

1, 3, 16, 120, 1140, 12972, 171216, 2571912, 43429680, 816108048, 16894168704, 381536713152, 9332214825024, 246215663789760, 6984603724315392, 211834855804295808, 6819603388970206464, 232454553855108173568

Offset: 0

Paul D. Hanna, Dec 27 2007, Jan 24 2008

nonn

			E.g.f.: A(x) = 1 + 3x + 16x^2/2! + 120x^3/3! + 1140x^4/4! + 12972x^5/5! +...

Cf. A027907; A005119 (variant).

{a(n)=if(n<0,0,if(n==0,1,(n-1)!*sum(i=1,n,(-1)^(i+1)*polcoeff((1+x+x^2)^(n-i+2),i+1)*a(n-i)/(n-i)!)))}

E.g.f. satisfies: A(x) = (1-x+x^2)^2/(1-2x)*A(x-x^2+x^3).

A145171 Triangle read by rows: left half of trinomial triangle (A027907) modulo 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Reikku Kulon, Oct 03 2008

Cf. A027907, A111808

cp's OEIS Frontend

A251686 G.f.: exp( Sum_{n>=1} A132303(n)/3 * x^n/n ), where A132303(n) = sum of the cubes of the trinomial coefficients in row n of triangle A027907.

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A027909 T(2n,n-1), T given by A027907.

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A027911 a(n) = T(2*n+1,n), with T given by A027907.

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A027912 T(2n-1,n-2), T given by A027907.

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A059781 Triangle T(n,k) giving exponent of power of 2 dividing entry (n,k) of trinomial triangle A027907.

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A059782 Triangle T(n,k) giving exponent of power of 3 dividing entry (n,k) of trinomial triangle A027907.

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A101617 The trinomial transform (A027907) gives powers of 3, while the trinomial transform of this sequence shift one place left gives powers of 5.

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A123934 Triangle T(n,k), 1<=k<=n :forms the odd-indexed trinomial coefficients (A027907).

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A136168 a(n) = (n-1)!*Sum_{i=1..n-1} (-1)^(i+1)*A027907(n-i+2,i+1)*a(n-i)/(n-i)! for n>0 with a(0)=1, where A027907 is the triangle of trinomial coefficients.

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A136168 a(n) = (n-1)!Sum_{i=1..n-1} (-1)^(i+1)A027907(n-i+2,i+1)*a(n-i)/(n-i)! for n>0 with a(0)=1, where A027907 is the triangle of trinomial coefficients.