A321600 -id:A321600 - cp's OEIS frontend

A322116 Main diagonal of triangle A321600; a(n) = A321600(n,n-1) for n >= 1.

2, 6, 26, 78, 242, 726, 2186, 6558, 19682, 59046, 177146, 531438, 1594322, 4782966, 14348906, 43046718, 129140162, 387420486, 1162261466, 3486784398, 10460353202, 31381059606, 94143178826, 282429536478, 847288609442, 2541865828326, 7625597484986, 22876792454958, 68630377364882, 205891132094646, 617673396283946, 1853020188851838, 5559060566555522, 16677181699666566

Offset: 1

Paul D. Hanna, Nov 26 2018

nonn

Triangle A321600 describes log( (1-y)*Sum_{n=-oo...+oo} (x^n + y)^n )/(1-y).

			G.f.: A(x) = 2*x + 6*x^2 + 26*x^3 + 78*x^4 + 242*x^5 + 726*x^6 + 2186*x^7 + 6558*x^8 + 19682*x^9 + 59046*x^10 + ...
L.g.f.: L(x)  =  log( (1-x)*(1-x^2)/(1-3*x) )  =  2*x + 6*x^2/2 + 26*x^3/3 + 78*x^4/4 + 242*x^5/5 + 726*x^6/6 + 2186*x^7/7 + 6558*x^8/8 + 19682*x^9/9 + 59046*x^10/10 + 177146*x^11/11 + ... + A321600(n,n-1)*x^n/n + ...
such that
exp(L(x)) = 1 + 2*x + 5*x^2 + 16*x^3 + 48*x^4 + 144*x^5 + 432*x^6 + 1296*x^7 + 3888*x^8 + 11664*x^9 + 34992*x^10 + 104976*x^11 + ... + A257970(n)*x^n + ...
exp(L(x)/2) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 44*x^5 + 122*x^6 + 342*x^7 + 966*x^8 + 2746*x^9 + 7846*x^10 + 22514*x^11 + 64836*x^12 + ... + A105696(n)*x^n + ...

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

Cf. A321600, A257970, A105696.

{a(n) = n*polcoeff( log((1 - x)*(1 - x^2)/(1 - 3*x +x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))

L.g.f.: log( (1 - x)*(1 - x^2)/(1 - 3*x) ).

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

A321601 G.f.: A(x,y) = Sum_{n=-oo...+oo} (x^n + y)^n = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + nk) y^k, written here as a rectangle of coefficients T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 4, 1, 2, 1, 9, 3, 1, 2, 1, 16, 6, 6, 1, 2, 1, 25, 10, 20, 5, 1, 2, 1, 36, 15, 50, 15, 8, 1, 2, 1, 49, 21, 105, 35, 35, 7, 1, 2, 1, 64, 28, 196, 70, 112, 28, 10, 1, 2, 1, 81, 36, 336, 126, 294, 84, 54, 9, 1, 2, 1, 100, 45, 540, 210, 672, 210, 210, 45, 12, 1, 2, 1, 121, 55, 825, 330, 1386, 462, 660, 165, 77, 11, 1, 2, 1, 144, 66, 1210, 495, 2640, 924, 1782, 495, 352, 66, 14, 1, 2, 1, 169, 78, 1716, 715, 4719, 1716, 4290, 1287, 1287, 286, 104, 13, 1, 2, 1, 196, 91, 2366, 1001, 8008, 3003, 9438, 3003, 4004, 1001, 546, 91, 16, 1, 2, 1, 225, 105, 3185, 1365, 13013, 5005, 19305, 6435, 11011, 3003, 2275, 455, 135, 15, 1

Offset: 0

Paul D. Hanna, Nov 25 2018

Related triangle A321600 describes log( (1-y)*Sum_{n=-oo...+oo} (x^n + y)^n ) / (1-y).

			G.f.: A(x,y)  =  Sum_{n=-oo...+oo} (x^n + y)^n  =  1/(1 - y) + x*(2) + x^2*(y) + x^3*(4*y^2) + x^4*(2 + 3*y^3) + x^5*(6*y^4) + x^6*(y + 5*y^5) + x^7*(8*y^6) + x^8*(9*y^2 + 7*y^7) + x^9*(2 + 10*y^8) + x^10*(6*y^3 + 9*y^9) + x^11*(12*y^10) + x^12*(y + 20*y^4 + 11*y^11) + x^13*(14*y^12) + x^14*(15*y^5 + 13*y^13) + x^15*(16*y^2 + 16*y^14) + x^16*(2 + 35*y^6 + 15*y^15) + x^17*(18*y^16) + x^18*(10*y^3 + 28*y^7 + 17*y^17) + x^19*(20*y^18) + x^20*(y + 54*y^8 + 19*y^19) + x^21*(50*y^4 + 22*y^20) + x^22*(45*y^9 + 21*y^21) + x^23*(24*y^22) + x^24*(25*y^2 + 35*y^5 + 77*y^10 + 23*y^23) + x^25*(2 + 26*y^24) + x^26*(66*y^11 + 25*y^25) + x^27*(112*y^6 + 28*y^26) + x^28*(15*y^3 + 104*y^12 + 27*y^27) + x^29*(30*y^28) + x^30*(y + 84*y^7 + 91*y^13 + 29*y^29) + x^31*(32*y^30) + x^32*(105*y^4 + 135*y^14 + 31*y^31) + x^33*(210*y^8 + 34*y^32) + x^34*(120*y^15 + 33*y^33) + x^35*(36*y^2 + 36*y^34) + x^36*(2 + 70*y^5 + 165*y^9 + 170*y^16 + 35*y^35) + ...
AS A RECTANGLE.
This sequence presents the coefficients of A(x,y) in the compact form
(*) A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + n*k) * y^k,
so that this table of coefficients T(n,k) of x^(n^2 + n*k) * y^k in A(x,y) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, ...;
2, 1, 9, 6, 20, 15, 35, 28, 54, 45, 77, 66, 104, ...;
2, 1, 16, 10, 50, 35, 112, 84, 210, 165, 352, 286, ...;
2, 1, 25, 15, 105, 70, 294, 210, 660, 495, 1287, 1001, ...;
2, 1, 36, 21, 196, 126, 672, 462, 1782, 1287, 4004, 3003, ...;
2, 1, 49, 28, 336, 210, 1386, 924, 4290, 3003, 11011, 8008, ...;
2, 1, 64, 36, 540, 330, 2640, 1716, 9438, 6435, 27456, ...;
2, 1, 81, 45, 825, 495, 4719, 3003, 19305, 12870, 63206, ...;
2, 1, 100, 55, 1210, 715, 8008, 5005, 37180, 24310, 136136, ...;
2, 1, 121, 66, 1716, 1001, 13013, 8008, 68068, 43758, 277134, ...;
...
where the g.f. of column k is (1+x)/( (1 - (-1)^k*x) * (1-x)^k ) for k >= 0.
Thus, for column k > 0,
if k odd: T(n,k) = binomial(n+k-1,k-1),
if k even: T(n,k) = binomial(n+k-1,k-1)*(2*n+k)/k.
...

Paul D. Hanna, Table of n, a(n) for n = 0..1325 of this rectangle as read by antidiagonals across rows 0..50

Cf. A321600 (log).

/* Table of coefficients in Sum_{n=-oo...+oo} (x^n + y)^n */
ROWS=12
{Q(n, k) = polcoeff(polcoeff( sum(m=-n-k, n+k, (x^m + y +O(x^(n+1)))^m ), n, x) +O(y^(k+1)), k, y)}
{T(n,k) = polcoeff( sum(j=0, k, Q(n^2 + n*j,j)*z^j +O(z^(k+1))), k, z)}
for(n=0, ROWS, for(k=0, ROWS, print1(T(n, k), ", ")); print(""))

/* Using binomial formula for terms T(n,k) */
ROWS=12
{T(n,k) = if(k==0, if(n==0,1,2),
if(k%2==1, binomial(n+k-1,k-1), binomial(n+k-1,k-1)*(2*n+k)/k))}
for(n=0, ROWS, for(k=0, ROWS, print1(T(n, k), ", ")); print(""))

G.f. A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + n*k) * y^k satisfies
(1) A(x,0) = theta_3(x), a Jacobi theta function.
(2) A(x,y) = Sum_{n=-oo..+oo} (x^n + y)^n.
(3) A(x,y) = Sum_{n=-oo..+oo} x^(n^2)/(1 - x^n*y)^(n+1).
(4) A(x,y) = theta_3(x) + Sum_{n>=0} Sum_{k>=0} x^(n*(n+2*k+1)) * y^(2*k+1) * [ binomial(n+2*k,2*k) + binomial(n+2*k+1,2*k+1)*(n+k+1)/(k+1) * x^n*y ].
FORMULAS FOR TERMS.
The o.g.f. of column k is (1+x)/( (1 - (-1)^k*x) * (1-x)^k ) for k >= 0.
Thus, for column k > 0,
if k odd: T(n,k) = binomial(n+k-1,k-1),
if k even: T(n,k) = binomial(n+k-1,k-1)*(2*n+k)/k.
Antidiagonal sums are [1, 3, 4, 8, 16, ... , 2^n, ...].

cp's OEIS Frontend

A322116 Main diagonal of triangle A321600; a(n) = A321600(n,n-1) for n >= 1.

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A321601 G.f.: A(x,y) = Sum_{n=-oo...+oo} (x^n + y)^n = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + nk) y^k, written here as a rectangle of coefficients T(n,k) read by antidiagonals.

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cp's OEIS Frontend

A322116 Main diagonal of triangle A321600; a(n) = A321600(n,n-1) for n >= 1.

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A321601 G.f.: A(x,y) = Sum_{n=-oo...+oo} (x^n + y)^n = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + n*k) * y^k, written here as a rectangle of coefficients T(n,k) read by antidiagonals.

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PARI

Formula

A321601 G.f.: A(x,y) = Sum_{n=-oo...+oo} (x^n + y)^n = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + nk) y^k, written here as a rectangle of coefficients T(n,k) read by antidiagonals.