A324958 -id:A324958 - cp's OEIS frontend

A324305 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-2} (n + ky + ny^2) for n > 1 with a single '1' in row 1.

1, 2, 0, 2, 9, 3, 18, 3, 9, 64, 48, 200, 96, 200, 48, 64, 625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625, 7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776, 117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649, 2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152

Offset: 1

Paul D. Hanna, Feb 28 2019

			E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2) and satisfies A(x,y) = x/(1 - y*A(x,y))^(1/y + y).
Explicitly,
A(x,y) = x + (2*y^2 + 2)*x^2/2! + (9*y^4 + 3*y^3 + 18*y^2 + 3*y + 9)*x^3/3! + (64*y^6 + 48*y^5 + 200*y^4 + 96*y^3 + 200*y^2 + 48*y + 64)*x^4/4! + (625*y^8 + 750*y^7 + 2775*y^6 + 2280*y^5 + 4300*y^4 + 2280*y^3 + 2775*y^2 + 750*y + 625)*x^5/5! + (7776*y^10 + 12960*y^9 + 46440*y^8 + 53640*y^7 + 100584*y^6 + 81360*y^5 + 100584*y^4 + 53640*y^3 + 46440*y^2 + 12960*y + 7776)*x^6/6! + (117649*y^12 + 252105*y^11 + 909979*y^10 + 1337700*y^9 + 2594501*y^8 + 2753415*y^7 + 3604342*y^6 + 2753415*y^5 + 2594501*y^4 + 1337700*y^3 + 909979*y^2 + 252105*y + 117649)*x^7/7! + ...
Setting y = 1 yields an o.g.f. of A006013:
A(x,y=1) = x + 2*x^2 + 7*x^3 + 30*x^4 + 143*x^5 + 728*x^6 + 3876*x^7 + 21318*x^8 + 120175*x^9 + ... + binomial(3*n-2,n-1)/n * x^n + ...
TRIANGLE.
This triangle of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, begins
1;
2, 0, 2;
9, 3, 18, 3, 9;
64, 48, 200, 96, 200, 48, 64;
625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776;
117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649;
2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152; ...
RELATED SERIES.
The e.g.f. may be defined by A(x,y) = Series_Reversion( x/G(x,y) )
where G(x,y) is the e.g.f. of A201949 and equals
G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2)
so that
G(x,y) = 1 + (1 + y^2)*x + (1 + y + 2*y^2 + y^3 + y^4)*x^2/2! + (1 + 3*y + 5*y^2 + 6*y^3 + 5*y^4 + 3*y^5 + y^6)*x^3/3! + (1 + 6*y + 15*y^2 + 24*y^3 + 28*y^4 + 24*y^5 + 15*y^6 + 6*y^7 + y^8)*x^4/4! + (1 + 10*y + 40*y^2 + 90*y^3 + 139*y^4 + 160*y^5 + 139*y^6 + 90*y^7 + 40*y^8 + 10*y^9 + y^10)*x^5/5! + ...
and G(x,y) = x / Series_Reversion( A(x,y) ).
RELATED TRIANGLE.
Triangle A201949 of coefficients in G(x,y) such that A(x/G(x,y),y) = x begins
1;
1, 0, 1;
1, 1, 2, 1, 1;
1, 3, 5, 6, 5, 3, 1;
1, 6, 15, 24, 28, 24, 15, 6, 1;
1, 10, 40, 90, 139, 160, 139, 90, 40, 10, 1;
1, 15, 91, 300, 629, 945, 1078, 945, 629, 300, 91, 15, 1; ...
where the g.f. of row n is Product_{k=0..n-1} (1 + k*y + y^2) for n >= 0.

Paul D. Hanna, Table of n, a(n) for n = 1..10000 terms in rows 1..100 of this triangle in flattened form.

Cf. A201949, A324304.

Cf. A324956, A324958.

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

/* A(x,y) = Series_Reversion(x/G(x,y)) where G(x,y) = e.g.f. A201949 */
{T(n,k) = my(G=1,A=x);
G = sum(m=0,n, x^m/m! * prod(j=0,m-1, 1 + j*y + y^2) +x*O(x^n));
A = serreverse(x/G);
n!*polcoeff(polcoeff(A,n,x),k,y)}
{for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}

GENERATING FUNCTIONS.
E.g.f.: A(x,y) = x/(1 - y*A(x,y))^(1/y + y).
E.g.f.: A(x,y) = Series_Reversion( x*(1 - x*y)^(1/y + y) ), wrt x.
E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2)
E.g.f.: A(x,y) = Series_Reversion( x/G(x,y) ) such that A(x/G(x,y),y) = x, where G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2) is the e.g.f. of A201949.
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: A(x,y=0) = -LambertW(-x) = x*exp(-LambertW(-x)).
E.g.f. at y = 1: A(x,y=1) = x*G(x)^2, where G = 1 + x*G(x)^3 is the g.f. of A001764.
FORMULAS INVOLVING TERMS.
Row sums: Sum_{k=0..2*n-2} T(n,k) = (3*n-2)!/(2*n-1)! for n >= 1.
T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
T(n,n-1) = A324304(n) for n >= 1.

A324960 Triangle of coefficients T(n,k) of y^k in Product_{k=0..n-1} (1 + (k+2)y + y^2), read by rows of terms k = 0..2n, for n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 1, 5, 8, 5, 1, 1, 9, 29, 42, 29, 9, 1, 1, 14, 75, 196, 268, 196, 75, 14, 1, 1, 20, 160, 660, 1519, 2000, 1519, 660, 160, 20, 1, 1, 27, 301, 1800, 6299, 13293, 17038, 13293, 6299, 1800, 301, 27, 1, 1, 35, 518, 4235, 21000, 65485, 129681, 162890, 129681, 65485, 21000, 4235, 518, 35, 1, 1, 44, 834, 8932, 59633, 258720, 740046, 1395504, 1725372, 1395504, 740046, 258720, 59633, 8932, 834, 44, 1, 1, 54, 1275, 17316, 149787, 863982, 3386879, 9054684, 16420458, 20044728, 16420458, 9054684, 3386879, 863982, 149787, 17316, 1275, 54, 1

Offset: 0

Paul D. Hanna, Mar 20 2019

			E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n} T(n,k)*y^k starts
A(x,y) = 1 + (y^2 + 2*y + 1)*x + (y^4 + 5*y^3 + 8*y^2 + 5*y + 1)*x^2/2! + (y^6 + 9*y^5 + 29*y^4 + 42*y^3 + 29*y^2 + 9*y + 1)*x^3/3! + (y^8 + 14*y^7 + 75*y^6 + 196*y^5 + 268*y^4 + 196*y^3 + 75*y^2 + 14*y + 1)*x^4/4! + (y^10 + 20*y^9 + 160*y^8 + 660*y^7 + 1519*y^6 + 2000*y^5 + 1519*y^4 + 660*y^3 + 160*y^2 + 20*y + 1)*x^5/5! + (y^12 + 27*y^11 + 301*y^10 + 1800*y^9 + 6299*y^8 + 13293*y^7 + 17038*y^6 + 13293*y^5 + 6299*y^4 + 1800*y^3 + 301*y^2 + 27*y + 1)*x^6/6! + (y^14 + 35*y^13 + 518*y^12 + 4235*y^11 + 21000*y^10 + 65485*y^9 + 129681*y^8 + 162890*y^7 + 129681*y^6 + 65485*y^5 + 21000*y^4 + 4235*y^3 + 518*y^2 + 35*y + 1)*x^7/7! + (y^16 + 44*y^15 + 834*y^14 + 8932*y^13 + 59633*y^12 + 258720*y^11 + 740046*y^10 + 1395504*y^9 + 1725372*y^8 + 1395504*y^7 + 740046*y^6 + 258720*y^5 + 59633*y^4 + 8932*y^3 + 834*y^2 + 44*y + 1)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y) begins:
  1;
  1, 2, 1;
  1, 5, 8, 5, 1;
  1, 9, 29, 42, 29, 9, 1;
  1, 14, 75, 196, 268, 196, 75, 14, 1;
  1, 20, 160, 660, 1519, 2000, 1519, 660, 160, 20, 1;
  1, 27, 301, 1800, 6299, 13293, 17038, 13293, 6299, 1800, 301, 27, 1;
  1, 35, 518, 4235, 21000, 65485, 129681, 162890, 129681, 65485, 21000, 4235, 518, 35, 1;
  1, 44, 834, 8932, 59633, 258720, 740046, 1395504, 1725372, 1395504, 740046, 258720, 59633, 8932, 834, 44, 1;
  1, 54, 1275, 17316, 149787, 863982, 3386879, 9054684, 16420458, 20044728, 16420458, 9054684, 3386879, 863982, 149787, 17316, 1275, 54, 1; ...

Cf. A324961, A324962.

Cf. A324958, A249790.

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

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

E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n} T(n,k)*y^k satisfies
(1) A(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + (k+2)*y + y^2).
(2) A(x,y) = 1/(1 - x*y)^((1+y)^2/y).
(3) x = Sum_{n>=1} (x/A(x,y))^n/n! * Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2).
Row sums are (n+3)!/3! for row n >= 0.

A324959 a(n) is the coefficient of y^(n-1) in Product_{k=0..n-2} (n + (2n + k)y + n*y^2), for n >= 1.

Original entry on oeis.org

1, 4, 60, 1584, 60460, 3029040, 188149822, 13957194496, 1204226253180, 118497226636800, 13098496404605964, 1607024046808249344, 216700840893719564902, 31857524157092264001280, 5071166437655033657264250, 868987068436739105218560000, 159492062728455524910446791332, 31215865935497559008870102593536, 6489956761227888786183691062551704, 1428394947783425181327275654594560000

Offset: 1

Paul D. Hanna, Mar 20 2019

nonn

a(n) = A324958(n,n-1) for n >= 1.

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Cf. A324958.

Table[If[n==1, 1, Coefficient[Expand[Product[(n + (2*n+k)*y + n*y^2), {k, 0, n-2}]], y^(n-1)]], {n, 1, 20}] (* Vaclav Kotesovec, Oct 30 2019 *)

{a(n) = polcoeff( prod(k=0, n-2, n + (2*n+k)*y + n*y^2 +y*O(y^n)), n-1, y)}
for(n=1, 25, print1(a(n), ", "))

a(n) ~ n! * c * 5^(5*n) / (2^(8*n) * n^2), where c = 1/(5*Pi*sqrt(10*log(5/4))) = 0.04261749831824651172873387091554100360007169830546206828545398795767677148... - Vaclav Kotesovec, Oct 30 2019, updated Mar 17 2024

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A324305 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-2} (n + ky + ny^2) for n > 1 with a single '1' in row 1.

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A324960 Triangle of coefficients T(n,k) of y^k in Product_{k=0..n-1} (1 + (k+2)y + y^2), read by rows of terms k = 0..2n, for n >= 0.

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A324959 a(n) is the coefficient of y^(n-1) in Product_{k=0..n-2} (n + (2n + k)y + n*y^2), for n >= 1.

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cp's OEIS Frontend

A324305 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a single '1' in row 1.

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A324960 Triangle of coefficients T(n,k) of y^k in Product_{k=0..n-1} (1 + (k+2)*y + y^2), read by rows of terms k = 0..2*n, for n >= 0.

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A324959 a(n) is the coefficient of y^(n-1) in Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2), for n >= 1.

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A324305 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-2} (n + ky + ny^2) for n > 1 with a single '1' in row 1.

A324960 Triangle of coefficients T(n,k) of y^k in Product_{k=0..n-1} (1 + (k+2)y + y^2), read by rows of terms k = 0..2n, for n >= 0.

A324959 a(n) is the coefficient of y^(n-1) in Product_{k=0..n-2} (n + (2n + k)y + n*y^2), for n >= 1.