A358499 -id:A358499 - cp's OEIS frontend

A136580 Row sums of triangle A136579.

1, 1, 3, 7, 27, 127, 747, 5167, 41067, 368047, 3669867, 40284847, 482671467, 6267305647, 87660962667, 1313941673647, 21010450850667, 357001369769647, 6423384156578667, 122002101778601647, 2439325392333218667

Offset: 0

Gary W. Adamson, Jan 09 2008

			a(4) = 27 = sum of row 4 terms, triangle A136579: (1 + 0 + 2 + 0 + 24) = 0! + 2! + 4!.
a(5) = 127 = sum of row 5 terms, triangle A136579: (0 + 1 + 0 + 6 + 0 + 120) = 1! + 3! + 5!
G.f. = 1 + x + 3*x^2 + 7*x^3 + 27*x^4 + 127*x^5 + 747*x^6 + 5167*x^7 + 41067*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..449
Eric Weisstein's World of Mathematics, Incomplete Gamma Function.
Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A136579, A000142, A153229, A058006.

Cf. A358498, A358499, A358500.

A136580 := proc(n)
    add( (n-2*i)!,i=0..floor(n/2) ) ;
end proc: # R. J. Mathar, Jun 04 2021

a[0] = 1; a[1] = 1; a[2] = 3; a[n_] := a[n] = n a[n-1] + a[n-2] - n a[n-3]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

G.f.: 2/(1-x^2)/G(0), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: Q(0)/(1-x^2), where Q(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
From Vladimir Reshetnikov, Oct 29 2015: (Start):
a(n) = (-1)^n*exp(1)*Gamma(0, 1)/2 - Re(Gamma(0, -1))*exp(-1)/2 + (n+2)!*((-1)^n*Re(Gamma(-n-2, -1))*exp(-1)-Gamma(-n-2, 1)*exp(1))/2, where Gamma(a, x) is the upper incomplete Gamma function.
D-finite with recurrence: a(0) = 1, a(1) = 1, a(2) = 3, a(n) = n*a(n-1) + a(n-2) - n*a(n-3).
E.g.f.: 1/(1-x) + (exp(x-1)*(Ei(1)-Ei(1-x)) + exp(1-x)*(Ei(x-1)-Ei(-1)))/2, where Ei(x) is the exponential integral.
a(n+1)-a(n) = A153229(n+2) = (-1)^(n+1)*A058006(n+1).
(End)
0 = a(n)*(+a(n+1) - a(n+2) - a(n+3) + a(n+4)) + a(n+1)*(+a(n+1) - a(n+2) - 2*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3) - a(n+4)) + a(n+3)*(+a(n+3)) for all n>=0. - Michael Somos, Oct 29 2015

A099948 Number of partitions of n such that the number of blocks is congruent to 3 mod 4.

Original entry on oeis.org

1, 6, 25, 90, 302, 994, 3487, 15210, 92489, 713988, 5979480, 50184316, 412595913, 3317961318, 26241631409, 205918294518, 1622545217510, 13045429410974, 109152638729439, 969395726250226, 9255388478615017, 94973500733767432, 1034488089509527120

Offset: 3

N. J. A. Sloane, Nov 12 2004

			a(11)=92489 because stirling2(11,3)+stirling2(11,7)+stirling2(11,11)=92489.

Alois P. Heinz, Table of n, a(n) for n = 3..500
M. Klazar, Bell numbers, their relatives and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.

Cf. A143817, A358499.

seq(sum(stirling2(n,3+4*k),k=0..(n-3)/4),n=3..26); # Emeric Deutsch, Dec 15 2004
# second Maple program:
with(combinat):
b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=3, 1, 0),
     `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1, irem(m+j, 4)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=3..30);  # Alois P. Heinz, Sep 17 2015

Table[Sum[StirlingS2[n, 3+4*k], {k, 0, (n-3)/4}], {n, 3, 26}] (* Jean-François Alcover, Feb 18 2016, after Emeric Deutsch *)

G.f.: sum(x^k/[(1-x)(1-2x)...(1-kx)], k=3 (mod 4)). - Emeric Deutsch, Dec 15 2004

More terms from Emeric Deutsch, Dec 15 2004

A358498 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)!.

Original entry on oeis.org

1, 1, 2, 7, 25, 122, 727, 5065, 40442, 363607, 3633865, 39957242, 479365207, 6230654665, 87218248442, 1308153733207, 20929020542665, 355774646344442, 6403681859461207, 121666029429374665, 2433257782822984442, 51097345853568901207, 1124122393807037054665

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A136580, A358499, A358500.

PARI
```
a(n) = sum(k=0, n\3, (n-3*k)!);
```

a(n) = n * a(n-1) + a(n-3) - n * a(n-4) for n > 3.

a(n) ~ n! * (1 + 1/n^3 + 3/n^4 + 7/n^5 + 16/n^6 + 46/n^7 + 203/n^8 + 1178/n^9 + 7242/n^10 + ...), for coefficients see A143817. - Vaclav Kotesovec, Nov 24 2022

A358500 a(n) = Sum_{k=0..floor(n/5)} (n-5*k)!.

Original entry on oeis.org

1, 1, 2, 6, 24, 121, 721, 5042, 40326, 362904, 3628921, 39917521, 479006642, 6227061126, 87178654104, 1307677996921, 20922829805521, 355687907102642, 6402379932789126, 121645187587486104, 2432903315854636921, 51090963094539245521, 1124001083465514782642

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A136580, A358498, A358499.

PARI
```
a(n) = sum(k=0, n\5, (n-5*k)!);
```

a(n) = n * a(n-1) + a(n-5) - n * a(n-6) for n > 5.

a(n) ~ n! * (1 + 1/n^5 + 10/n^6 + 65/n^7 + 350/n^8 + 1701/n^9 + 7771/n^10 + 34150/n^11 + 146905/n^12 + ...), the coefficients are Sum_{j=0..(k-4)/5} Stirling2(k,5*j+4). - Vaclav Kotesovec, Nov 24 2022

A358609 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * (n-4*k)!.

Original entry on oeis.org

1, 1, 2, 6, 23, 119, 718, 5034, 40297, 362761, 3628082, 39911766, 478961303, 6226658039, 87174663118, 1307634456234, 20922310926697, 355681201437961, 6402286531064882, 121643792774375766, 2432881085865713303, 51090586490508002039, 1123994325491076615118

Offset: 0

Seiichi Manyama, Nov 23 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A358607, A358608, A358611.

Cf. A358499.

PARI
```
a(n) = sum(k=0, n\4, (-1)^k*(n-4*k)!);
```

a(n) = n * a(n-1) - a(n-4) + n * a(n-5) for n > 4.

a(n) ~ n! * (1 - 1/n^4 - 6/n^5 - 25/n^6 - 90/n^7 - 300/n^8 - 938/n^9 - 2563/n^10 - ...). - Vaclav Kotesovec, Nov 25 2022

cp's OEIS Frontend

A136580 Row sums of triangle A136579.

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A099948 Number of partitions of n such that the number of blocks is congruent to 3 mod 4.

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A358498 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)!.

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A358500 a(n) = Sum_{k=0..floor(n/5)} (n-5*k)!.

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A358609 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * (n-4*k)!.

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