A084949 -id:A084949 - cp's OEIS frontend

A084948 a(n) = Product_{i=0..n-1} (8*i+2).

1, 2, 20, 360, 9360, 318240, 13366080, 668304000, 38761632000, 2558267712000, 189311810688000, 15523568476416000, 1397121162877440000, 136917873961989120000, 14513294639970846720000, 1654515588956676526080000, 201850901852714536181760000, 26240617240852889703628800000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

G. C. Greubel, Table of n, a(n) for n = 0..330

Cf. A000079, A000142, A000165, A008544, A001813, A047055, A047657, A048994, A084943, A084947, A084949.

List([0..20], n-> Product([0..n-1], k-> 8*k+2) ); # G. C. Greubel, Aug 18 2019

[1] cat [(&*[8*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019

a := n->product(8*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Table[8^n*Pochhammer[1/4, n], {n,0,20}] (* G. C. Greubel, Aug 18 2019 *)

vector(20, n, n--; prod(k=0, n-1, 8*k+2)) \\ G. C. Greubel, Aug 18 2019

[product(8*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

a(n) = A084943(n)/A000142(n)*A000079(n) = 8^n*Pochhammer(1/4, n) = 1/2*Gamma(n+1/4)*sqrt(2)*Gamma(3/4)*8^n/Pi.
a(n) = (-6)^n*Sum_{k=0..n} (4/3)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+2)/(2*x*(8*k+2) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 8*x)^(1/4).
a(n) ~ sqrt(2*Pi)*8^n*n^n/(exp(n)*n^(1/4)*Gamma(1/4)). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/8^6)^(1/8)*(Gamma(1/4) - Gamma(1/4, 1/8)). - Amiram Eldar, Dec 20 2022

A257608 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 9*x + 1.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 219, 219, 1, 1, 2218, 8322, 2218, 1, 1, 22217, 220222, 220222, 22217, 1, 1, 222216, 5006247, 12332432, 5006247, 222216, 1, 1, 2222215, 105340629, 530539235, 530539235, 105340629, 2222215, 1, 1, 22222214, 2123693776, 19700767514, 39259903390, 19700767514, 2123693776, 22222214, 1

Offset: 0

Dale Gerdemann, May 03 2015

			Triangle begins as:
  1;
  1,       1;
  1,      20,         1;
  1,     219,       219,         1;
  1,    2218,      8322,      2218,         1;
  1,   22217,    220222,    220222,     22217,         1;
  1,  222216,   5006247,  12332432,   5006247,    222216,       1;
  1, 2222215, 105340629, 530539235, 530539235, 105340629, 2222215, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_9(n,k).

Cf. A084949 (row sums), A257619.
Cf. A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A144431, A167884
Similar sequences listed in A256890.

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,9,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

def T(n,k,a,b): # A257608
    if (k<0 or k>n): return 0
    elif (k==0 or k==n): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,9,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

T(n, k) = t(n-k, k), where t(n,k) = f(k)*t(n-1, k) + f(n)*t(n, k-1), and f(n) = 9*n + 1.
Sum_{k=0..n} T(n, k) = A084949(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = T(n, n) = 1, a = 9, and b = 1. - G. C. Greubel, Mar 20 2022

cp's OEIS Frontend

A084948 a(n) = Product_{i=0..n-1} (8*i+2).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A257608 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 9*x + 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

cp's OEIS Frontend

A084948 a(n) = Product_{i=0..n-1} (8*i+2).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A257608 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A257608 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 9*x + 1.