A123680 -id:A123680 - cp's OEIS frontend

A123681 a(n) = (1/(n+1)) * Sum_{k=0..n} C(n+k-1,k)*k! = A123680(n)/(n+1).

1, 1, 3, 19, 197, 2841, 52327, 1171871, 30899529, 937529317, 32173291931, 1232093935227, 52088478142861, 2409578607253169, 121067200114483407, 6565538372492694871, 382234458749760846737, 23777755561583494209981

Offset: 0

Paul D. Hanna, Oct 05 2006

nonn

			a(n) = (Pochhammer(n, n + 1)*subfactorial(-2*n - 1) + (-1)^n*subfactorial(-n))/(n+1) where subfactorial(n) = exp(-1)*Gamma(n + 1, -1). - _Peter Luschny_, Oct 18 2017

G. C. Greubel, Table of n, a(n) for n = 0..365

Cf. A123680.

subfactorial := n -> simplify(exp(-1)*GAMMA(n+1,-1)):
a := n -> (pochhammer(n,n+1)*subfactorial(-2*n-1)+(-1)^n*subfactorial(-n))/(n+1):
seq(simplify(evalc(a(n))), n=0..17); # Peter Luschny, Oct 18 2017

Table[1/(n+1) Sum[Binomial[n+k-1,k]k!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Dec 14 2012 *)

a(n)=sum(k=0,n,binomial(n+k-1,k)*k!)/(n+1)

a(n) = (1/(n+1)) * Sum_{k=0..n} k! * [x^k] 1/(1-x)^n.

a(n) ~ 2^(2*n - 1/2) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 27 2017

Definition corrected by Harvey P. Dale, Dec 14 2012

A124320 Triangle read by rows: T(n,k) = k!*binomial(n+k-1,k) (n >= 0, 0 <= k <= n), rising factorial power, Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 1, 3, 12, 60, 1, 4, 20, 120, 840, 1, 5, 30, 210, 1680, 15120, 1, 6, 42, 336, 3024, 30240, 332640, 1, 7, 56, 504, 5040, 55440, 665280, 8648640, 1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200, 1, 9, 90, 990, 11880, 154440, 2162160, 32432400, 518918400, 8821612800

Offset: 0

Emeric Deutsch, Oct 26 2006

This is the Pochhammer function which is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1. Also known as the rising factorial power. - Peter Luschny, Jan 09 2011

			Triangle starts:
[0]  1;
[1]  1, 1;
[2]  1, 2,   6;
[3]  1, 3,  12,  60;
[4]  1, 4,  20, 120,  840;
[5]  1, 5,  30, 210, 1680, 15120;
[6]  1, 6,  42, 336, 3024, 30240, 332640;
[7]  1, 7,  56, 504, 5040, 55440, 665280, 8648640;
Array starts:
[0] 1,  1,   6,   60,   840,   15120,   332640,   8648640, ... A000407
[1] 1,  2,  12,  120,  1680,   30240,   665280,  17297280, ... A001813
[2] 1,  3,  20,  210,  3024,   55440,  1235520,  32432400, ... A006963
[3] 1,  4,  30,  336,  5040,   95040,  2162160,  57657600, ... A001761
[4] 1,  5,  42,  504,  7920,  154440,  3603600,  98017920, ... A102693
[5] 1,  6,  56,  720, 11880,  240240,  5765760, 160392960, ... A093197
[6] 1,  7,  72,  990, 17160,  360360,  8910720, 253955520, ... A203473
[7] 1,  8,  90, 1320, 24024,  524160, 13366080, 390700800, ...
[8] 1,  9, 110, 1716, 32760,  742560, 19535040, 586051200, ...
[9] 1, 10, 132, 2184, 43680, 1028160, 27907200, 859541760, ...

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
NIST Digital Library of Mathematical Functions, Pochhammer's Symbol

Rows of array: A000407, A001813, A006963, A001761, A102693, A093197, A203473.

Cf. A123680 (row sums), A352601 (array main diagonal), A123680, A068424 (falling factorial power).

T:=proc(n,k) if k<=n then binomial(n+k-1,k)*k! else 0 fi end: for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
A124320 := (n,k)-> `if`(n=0 and k=0,1,pochhammer(n,k)); seq(print(seq(A124320(n,k),k=0..n)),n=0..5); # Peter Luschny, Jan 09 2011

Table[Pochhammer[n,k], {n,0,5},{k,0,n}]//Flatten (* Peter Luschny, Jan 09 2011 *)

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, (n+k-1)!/(n-1)!), ", "))) \\ G. C. Greubel, Nov 19 2017

for n in (0..5) : [rising_factorial(n, k) for k in (0..n)] # Peter Luschny, Jan 09 2011

T(n,k) = GAMMA(n+k)/GAMMA(n) for n>0. - Peter Luschny, Jan 09 2011

cp's OEIS Frontend

A123681 a(n) = (1/(n+1)) * Sum_{k=0..n} C(n+k-1,k)*k! = A123680(n)/(n+1).

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