A030297 -id:A030297 - cp's OEIS frontend

A121682 Triangle read by rows: T(i,j) = (T(i-1,j) + i)*i.

1, 6, 4, 27, 21, 9, 124, 100, 52, 16, 645, 525, 285, 105, 25, 3906, 3186, 1746, 666, 186, 36, 27391, 22351, 12271, 4711, 1351, 301, 49, 219192, 178872, 98232, 37752, 10872, 2472, 456, 64, 1972809, 1609929, 884169, 339849, 97929, 22329, 4185, 657, 81, 19728190, 16099390, 8841790, 3398590, 979390, 223390, 41950, 6670, 910, 100

Offset: 1

Thomas Wieder, Aug 15 2006

The first column is A030297 = a(n) = n*(n+a(n-1)). The main diagonal are the squares A000290 = n^2. The first lower diagonal (6,21,52,...) is A069778 = q-factorial numbers 3!_q. See also A121662.

			Triangle begins:
      1
      6     4
     27    21     9
    124   100    52   16
    645   525   285  105  25
   3906  3186  1746  666  186  36
  27391 22351 12271 4711 1351 301 49
  ...

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Cf. A030297, A000290, A069778, A121662.

Row sums give A337001.

T:= proc(i, j) option remember;
      `if`(j<1 or j>i, 0, (T(i-1, j)+i)*i)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 22 2022

T[n_, k_] /; 1 <= k <= n := T[n, k] = (T[n-1, k]+n)*n;
T[, ] = 0;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2022 *)

def T(i, j): return (T(i-1, j)+i)*i if 1 <= j <= i else 0
print([T(r, c) for r in range(1, 11) for c in range(1, r+1)]) # Michael S. Branicky, Jun 22 2022

Edited by N. J. A. Sloane, Sep 15 2006

Formula in name corrected by Alois P. Heinz, Jun 22 2022

A107283 E.g.f. exp(x)*(x^2+x+2)/(1-x).

Original entry on oeis.org

2, 5, 16, 59, 254, 1297, 7820, 54791, 438394, 3945629, 39456392, 434020435, 5208245366, 67707189929, 947900659204, 14218509888287, 227496158212850, 3867434689618741, 69613824413137664, 1322662663849615979, 26453253276992319982, 555518318816838720065

Offset: 0

Creighton Dement, May 19 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A030297.

CoefficientList[Series[E^x*(x^2+x+2)/(1-x), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 17 2012 *)

x='x+O('x^66); Vec(serlaplace(exp(x)*(x^2+x+2)/(1-x))) \\ Joerg Arndt, May 15 2013

Recurrence: (4*n+3)*a(n) = (4*n^2 + 7*n + 20)*a(n-1) - (4*n^2 + 16*n - 11)*a(n-2) + 9*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ 4*e*n!. - Vaclav Kotesovec, Oct 17 2012

A371831 a(n) = numerator(Sum_{k=1..n} k^2/k!).

Original entry on oeis.org

0, 1, 3, 9, 31, 43, 217, 3913, 9133, 73067, 1972819, 6576067, 24112247, 372017527, 1612075951, 157983443203, 7109254944151, 37916026368811, 644572448269793, 34806912206568841, 2422459091299663, 7775794614048301, 277759159408419360043, 2036900502328408640323, 46848711553553398727437

Offset: 0

Stefano Spezia, Apr 07 2024

Eric Weisstein's World of Mathematics, Incomplete Gamma Function.
Wikipedia, Incomplete gamma function.

Cf. A019762, A030297, A371832.

Cf. A014973, A092043.

a[n_]:=Numerator[(2(E*Gamma[n+1,1]-1)-n)/n!]; Array[a,25,0]

a(n) = numerator(sum(k=1, n, k^2/k!)); \\ Michel Marcus, Apr 07 2024

a(n) = numerator((2*(e*Gamma(n+1, 1) - 1) - n)/n!).
a(n) = numerator(A030297(n)/n!).
Limit_{n->oo} a(n)/A371832(n) = 2*e = A019762.

A371832 a(n) = denominator(Sum_{k=1..n} k^2/k!).

Original entry on oeis.org

1, 1, 1, 2, 6, 8, 40, 720, 1680, 13440, 362880, 1209600, 4435200, 68428800, 296524800, 29059430400, 1307674368000, 6974263296000, 118562476032000, 6402373705728000, 445586448384000, 1430277488640000, 51090942171709440000, 374666909259202560000, 8617338912961658880000

Offset: 0

Stefano Spezia, Apr 07 2024

Eric Weisstein's World of Mathematics, Incomplete Gamma Function.
Wikipedia, Incomplete gamma function.

Cf. A019762, A030297, A371831.

Cf. A014973, A092043.

a[n_]:=Denominator[(2(E*Gamma[n+1,1]-1)-n)/n!]; Array[a,25,0]

a(n) = denominator(sum(k=1, n, k^2/k!)); \\ Michel Marcus, Apr 07 2024

a(n) = denominator((2*(e*Gamma(n+1, 1) - 1) - n)/n!).
a(n) = denominator(A030297(n)/n!).
Limit_{n->oo} A371831(n)/a(n) = 2*e = A019762.

A348311 a(n) = n! * Sum_{k=1..n} (-1)^k * (k-2) / (k-1)!.

Original entry on oeis.org

0, 1, 2, 3, 20, 85, 534, 3703, 29672, 266985, 2669930, 29369131, 352429692, 4581585853, 64142202110, 962133031455, 15394128503504, 261700184559313, 4710603322067922, 89501463119290195, 1790029262385804260, 37590614510101889061, 826993519222241559782

Offset: 0

Ilya Gutkovskiy, Oct 11 2021

nonn

Cf. A000166, A000240, A030297, A053817, A180189, A212291.

Table[n! Sum[(-1)^k (k - 2)/(k - 1)!, {k, 1, n}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[x (1 + x) Exp[-x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n!*sum(k=1, n, (-1)^k * (k-2) / (k-1)!); \\ Michel Marcus, Oct 20 2021

E.g.f.: x * (1 + x) * exp(-x) / (1 - x).
a(0) = 0; a(n) = n * (a(n-1) + (-1)^n * (n-2)).
a(n) = n * (2 * A000166(n-1) + (-1)^n).

A374844 a(n) = n! * Sum_{k=1..n} k^k / k!.

Original entry on oeis.org

0, 1, 6, 45, 436, 5305, 78486, 1372945, 27760776, 637267473, 16372674730, 465411092641, 14501033559948, 491388542871577, 17991446425760094, 707765586767260785, 29770993461985724176, 1333347150740094075169, 63346656788618230928466

Offset: 0

Seiichi Manyama, Jul 22 2024

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007526, A030297, A337001, A337002.

Cf. A000142, A277506.

a:= proc(n) a(n):= n*a(n-1) + n^n end: a(0):= 0:
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 22 2024

PARI
```
a(n) = n!*sum(k=1, n, k^k/k!);
```

a(0) = 0; a(n) = n*a(n-1) + n^n.
a(n) = A277506(n) - n!.
E.g.f.: -1/( (1 + 1/LambertW(-x)) * (1 - x) ).
a(n) ~ n^n / (1 - exp(-1)). - Vaclav Kotesovec, Jul 22 2024

cp's OEIS Frontend

A121682 Triangle read by rows: T(i,j) = (T(i-1,j) + i)*i.

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A107283 E.g.f. exp(x)*(x^2+x+2)/(1-x).

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A371831 a(n) = numerator(Sum_{k=1..n} k^2/k!).

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A371832 a(n) = denominator(Sum_{k=1..n} k^2/k!).

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A348311 a(n) = n! * Sum_{k=1..n} (-1)^k * (k-2) / (k-1)!.

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Mathematica

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Formula

A374844 a(n) = n! * Sum_{k=1..n} k^k / k!.

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