A007778 -id:A007778 - cp's OEIS frontend

A345632 Sum of terms of even index in the binomial decomposition of n^(n-1).

1, 1, 5, 28, 353, 3376, 66637, 908608, 24405761, 432891136, 14712104501, 321504185344, 13218256749601, 343360783937536, 16565151205544957, 498676704524517376, 27614800115689879553, 945381827279671853056, 59095217374989483261925, 2267322327322331161821184, 157904201452248753415276001

Offset: 1

Olivier Gérard, Jun 21 2021

When writing n^(n-1) (A000169) as a sum of powers of n using the binomial theorem, one can separately sum the even and the odd powers of n. This is the even part.

Cf. A345633 (odd part).
Cf. A062024, A302583.
Cf. A000169, A007778, A092364, A081131.

Table[Plus @@ Table[(n-1)^(2 k) Binomial[n-1, 2 k], {k, 0, Floor[n/2]}], {n, 1, 21}]

a(n+1) = Sum_{k=0..floor(n/2)} n^(2k) binomial(n, 2k).

a(n+1) = ((1 - n)^n + (1 + n)^n)/2. - Stefano Spezia, Jun 21 2021

A345633 Sum of terms of odd index in the binomial decomposition of n^(n-1).

Original entry on oeis.org

0, 1, 4, 36, 272, 4400, 51012, 1188544, 18640960, 567108864, 11225320100, 421504185344, 10079828372880, 450353989316608, 12627774819845668, 654244800082329600, 21046391759976988928, 1240529732459024678912, 45032132922921758270916, 2975557672677668838178816

Offset: 1

Olivier Gérard, Jun 21 2021

When writing n^(n-1) (A000169) as a sum of powers of n using the binomial theorem, one can separately sum the even and the odd powers of n. This is the odd part. See the Formula section.

Cf. A345632 (even part).
Cf. A062024, A302583.
Cf. A000169, A007778, A092364, A081131.

Table[Plus @@ Table[(n - 1)^(2 k + 1) Binomial[n - 1, 2 k + 1], {k, 0, Floor[(n - 1)/2]}], {n, 1, 21}]

a(n+1) = Sum_{k=0..floor((n-1)/2)} n^(2k+1)*binomial(n, 2k+1).

a(n+1) = ((1 + n)^n - (1 - n)^n)/2.

A346963 Decimal expansion of Integral_{x=-1/e..0} LambertW(x)*LambertW(-1,x) dx.

Original entry on oeis.org

2, 1, 6, 5, 7, 7, 7, 7, 0, 4, 3, 6, 0, 0, 7, 2, 7, 7, 3, 5, 9, 0, 2, 4, 9, 2, 0, 0, 6, 0, 7, 3, 8, 3, 3, 1, 6, 9, 8, 7, 3, 5, 5, 8, 2, 2, 5, 5, 3, 5, 5, 6, 9, 3, 2, 7, 2, 3, 3, 1, 4, 4, 1, 6, 9, 4, 0, 9, 9, 6, 2, 2, 2, 7, 2, 2, 3, 6, 8, 0, 9, 8, 4, 8, 3, 0, 3, 8, 5, 9, 2, 2, 4, 8, 5, 2, 1, 1, 1, 1, 5, 7, 5, 4, 3

Offset: 0

Gleb Koloskov, Aug 09 2021

			0.216577770436007277359024920060738331698735582255355693272331441694...

Cf. A000169, A007778, A068985, A135003, A135011, A346962.

evalf(Integrate(LambertW(x)*LambertW(-1, x), x = -exp(-1)..0), 120); # Vaclav Kotesovec, Aug 23 2021

N[Integrate[LambertW[x]*LambertW[-1,x],{x,-1/E,0}],120]

11*exp(-1)-4+sumpos(n=1,(1/(1+1./n))^n/(n*(n+1)^2))

Equals Integral_{x=-1/e..0} LambertW(x)*LambertW(-1,x) dx.
Equals (3/e) - 1 + Sum_{n>0} (n^(n-1)/(n+1)^(n+2))*(Gamma(n+2,n+1)/Gamma(n+2)).
Equals (11/e)-4+Sum_{n>0} n^(n-1)/(n+1)^(n+2) = A135011-4+Sum_{n>0} A000169(n)/A007778(n+1).

A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

Original entry on oeis.org

1, -1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 5, 1, 0, 1, 20, 21, 7, 1, 0, 1, 48, 81, 40, 9, 1, 0, 1, 112, 297, 208, 65, 11, 1, 0, 1, 256, 1053, 1024, 425, 96, 13, 1, 0, 1, 576, 3645, 4864, 2625, 756, 133, 15, 1, 0, 1, 1280, 12393, 22528, 15625, 5616, 1225, 176, 17, 1

Offset: 0

Stefano Spezia, Jan 31 2025

			The array begins as:
   1, 1,   1,    1,     1,     1, ...
  -1, 1,   3,    5,     7,     9, ...
   0, 1,   8,   21,    40,    65, ...
   0, 1,  20,   81,   208,   425, ...
   0, 1,  48,  297,  1024,  2625, ...
   0, 1, 112, 1053,  4864, 15625, ...
   0, 1, 256, 3645, 22528, 90625, ...
   ...

Cf. A000012 (k=1 or n=0), A000567 (n=2), A001792 (k=2), A007778, A060747 (n=1), A081038 (k=3), A081039 (k=4), A081040 (k=5), A081041 (k=6), A081042 (k=7), A081043 (k=8), A081044 (k=9), A081045 (k=10), A103532, A154955, A380748 (antidiagonal sums).

A[0,0]:=1; A[1,0]:=-1; A[n_,k_]:=((k-1)*n+k)k^(n-1); Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1-x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+(k-1)*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n,k) = ((k - 1)*n + k)*k^(n-1) with A(0,0) = 1.
A(n,k) = n! * [x^n] exp(k*x)*(1 + (k - 1)*x).
A(n,0) = A154955(n+1).
A(3,n) = A103532(n-1) for n > 0.
A(n,n) = A007778(n) for n > 0.

A037184 Functional digraphs with 1 node not in the image.

Original entry on oeis.org

6, 48, 420, 3840, 38010, 407904, 4739112, 59405760, 800309070, 11542072080, 177536452236, 2902434644928, 50271450894210, 919799178083520, 17730187503894480, 359183503307677824, 7629999730919490582, 169605272611054951920, 3937616869933172881140

Offset: 3

Erich Friedman

Comment from Christian G. Bower: Labeled graphs where every node has outdegree 1, one node has indegree 0, one node has indegree 2, all other nodes have indegree 1 and there are no loops of length 1.

A000166 (derangements) gives number of functional digraphs with all nodes in the range, A007778 gives number of functional digraphs with n nodes.

a(n) = n * ( (n-2)*a(n-1) + (n-1)*a(n-2) ) / (n-3).

E.g.f.: x^3 / ((1-x)^3 * e^x).

A090650 n^(n+6).

Original entry on oeis.org

1, 256, 19683, 1048576, 48828125, 2176782336, 96889010407, 4398046511104, 205891132094649, 10000000000000000, 505447028499293771, 26623333280885243904, 1461920290375446110677, 83668255425284801560576

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Dec 13 2003

nonn

Cf. A000312, A007778, A008788, A008789, A008790, A008791.

a:=n->mul(n,k=-5..n):seq(a(n),n=1..14); # Zerinvary Lajos, Jan 26 2008

Table[n^(n+6),{n,0,40}](* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

A122606 n^(n+1) mod 7.

Original entry on oeis.org

0, 1, 1, 4, 2, 1, 6, 0, 1, 2, 5, 1, 5, 1, 0, 1, 4, 1, 4, 4, 6, 0, 1, 1, 3, 2, 6, 1, 0, 1, 2, 2, 1, 2, 6, 0, 1, 4, 6, 4, 3, 1, 0, 1, 1, 4, 2, 1, 6, 0, 1, 2, 5, 1, 5, 1, 0, 1, 4, 1, 4, 4, 6, 0, 1, 1, 3, 2, 6, 1, 0, 1, 2, 2, 1, 2, 6, 0, 1, 4, 6, 4, 3, 1, 0, 1, 1, 4, 2, 1, 6, 0, 1, 2, 5, 1, 5, 1, 0, 1, 4, 1, 4, 4, 6

Offset: 0

Zak Seidov, Sep 24 2006, Sep 25 2006

nonn

Cf. A053879, A070472, A007778.

Table[PowerMod[n,n+1,7],{n,0,120}] (* Harvey P. Dale, May 24 2012 *)

a(n) = n^(n+1) mod 7 = A007778(n) mod 7.

Periodic with cycle consisting of first 42 terms: {0,1,1,4,2,1,6,0,1,2,5,1,5,1,0,1,4,1,4,4,6,0,1,1,3,2,6,1,0,1,2,2,1,2,6,0,1,4,6,4,3,1}.

A134574 Array, a(n,k) = total size of all n-length words on a k-letter alphabet, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 24, 18, 4, 5, 64, 81, 32, 5, 6, 160, 324, 192, 50, 6, 7, 384, 1215, 1024, 375, 72, 7, 8, 896, 4374, 5120, 2500, 648, 98, 8, 9, 2048, 15309, 24576, 15625, 5184, 1029, 128, 9, 10, 4608, 52488, 114688, 93750, 38880, 9604, 1536, 162, 10

Offset: 1

Ross La Haye, Jan 22 2008

			a(2,2) = 8 because there are 2^2 = 4 2-length words on a 2 letter alphabet, each of size 2 and 2*4 = 8.
Array begins:
==================================================================
n\k|  1     2       3        4         5         6          7  ...
---|--------------------------------------------------------------
1  |  1     2       3        4         5         6          7  ...
2  |  2     8      18       32        50        72         98  ...
3  |  3    24      81      192       375       648       1029  ...
4  |  4    64     324     1024      2500      5184       9604  ...
5  |  5   160    1215     5120     15625     38880      84035  ...
6  |  6   384    4374    24576     93750    279936     705894  ...
7  |  7   896   15309   114688    546875   1959552    5764801  ...
8  |  8  2048   52488   524288   3125000  13436928   46118408  ...
9  |  9  4608  177147  2359296  17578125  90699264  363182463  ...
... - _Franck Maminirina Ramaharo_, Aug 07 2018

Cf. a(n, 1) = a(1, k) = A000027(n); a(n, 2) = A036289(n); a(n, 3) = A036290(n); a(n, 4) = A018215(n); a(n, 5) = A036291(n); a(n, 6) = A036292(n); a(n, 7) = A036293(n); a(n, 8) = A036294(n); a(2, k) = A001105(k); a(3, k) = A117642(k); a(n, n) = A007778(n); a(n, n+1) = A066274(n+1): sum[a(i-1, n-i+1), {i, 1, n}] = A062807(n).

t[n_, k_] := Sum[k^n, {j, n}]; Table[ t[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Robert G. Wilson v, Aug 07 2018 *)

a(n,k) = n*k^n.
O.g.f. (by columns): (k*x)/(-1+k*x)^2.
E.g.f. (by columns): k*x*exp(k*x).
a(n,k) = Sum[k^n,{j,1,n}] = n*Sum[C(n,m)*(k-1)^m,{m,0,n}]. - Ross La Haye, Jan 26 2008

A173249 Partial sums of A000272.

Original entry on oeis.org

1, 2, 3, 6, 22, 147, 1443, 18250, 280394, 5063363, 105063363, 2463011054, 64380375278, 1856540769315, 58550453144611, 2004745521503986, 74062339559431922, 2936485391069247715, 124376016487663499491

Offset: 0

Jonathan Vos Post, Feb 13 2010

Partial sums of number of trees on n labeled nodes. The subsequence of primes in this sequence begin: 2, 58550453144611, no more through a(30).

			a(19) = 1 + 1 + 1 + 3 + 16 + 125 + 1296 + 16807 + 262144 + 4782969 + 100000000 + 2357947691 + 61917364224 + 1792160394037 + 56693912375296 + 1946195068359375 + 72057594037927936 + 2862423051509815793 + 121439531096594251776 + 5480386857784802185939.

Cf. A000055, A000169, A000312, A007778, A007830, A008785-A008791, A033842, A000272, A036361, A036362, A036506, A000055, A054581, A097170

a(n) = SUM[i=0..n] A000272(i) = SUM[i=0..n] i^(i-2).

A281620 Triangle read by rows: Poincaré polynomials of orbifold of Fermat hypersurfaces.

Original entry on oeis.org

1, 7, 1, 67, 13, 1, 821, 181, 21, 1, 12281, 2906, 406, 31, 1, 217015, 53719, 8359, 799, 43, 1, 4424071, 1129899, 188707, 20637, 1429, 57, 1, 102207817, 26710345, 4690249, 561481, 45385, 2377, 73, 1, 2639010709, 701908264, 127951984, 16349374, 1469026, 91216, 3736, 91, 1

Offset: 2

F. Chapoton, Jan 25 2017

			The first few polynomials are 1; q + 7; q^2 + 13*q + 67; ...
Triangle begins:
        1;
        7,       1;
       67,      13,      1;
      821,     181,     21,     1;
    12281,    2906,    406,    31,    1;
   217015,   53719,   8359,   799,   43,  1;
  4424071, 1129899, 188707, 20637, 1429, 57, 1;
  ...

So Okada, Homological mirror symmetry of Fermat polynomials, arxiv:0910.2014 [math.AG], 2009-2010.

Row sums give A007778(n-1), alternating row sums are A281596.

T:= n-> (p-> seq(coeff(p, q, i), i=0..n-2))(add(add(n^j*
         binomial(n, j)*(-1)^(i+n+j)*binomial(n-2-j+1, i+1)*
         q^i, i=0..n-1-j), j=0..n-1)):
seq(T(n), n=2..10);  # Alois P. Heinz, Jan 25 2017
# Alternatively:
t := n -> factor(((-n-x+1)^n+(x-1)*(1-n)^n-(-n)^n*x)*(-1)^n/((x-1)*x)):
seq(seq(coeff(t(n),x,k),k=0..n-2),n=2..10); # Peter Luschny, Jan 26 2017

T[n_] := ((-n-x+1)^n+(x-1)(1-n)^n-(-n)^n x) (-1)^n/((x-1) x); Table[CoefficientList[T[n],x],{n,2,10}] // Flatten (* Peter Luschny, Jan 26 2017 *)

def fermat(n):
    q = polygen(ZZ, 'q')
    return sum(n**j * binomial(n, j) * (-1)**(i + n + j) *
               binomial(n - 2 - j + 1, i + 1) * q**i
               for j in range(n - 1)
               for i in range(n - 1 - j))

# Alternatively:
def A281620_row(n):
    x = polygen(ZZ, 'x')
    p = (((-n-x+1)^n + (x-1)*(1-n)^n - (-n)^n*x)*(-1)^n)//((x-1)*x)
    return p.list()
for n in (2..10): print(A281620_row(n)) # Peter Luschny, Jan 26 2017

The formula given by Okada needs to be corrected as follows:
Sum_{j=0..n-1} Sum_{i=0..n-1-j} n^j * binomial(n,j) * (-1)^(i+n+j) * binomial(n-2-j+1,i+1) * q^i.
From Peter Luschny, Jan 26 2017: (Start)
T(n,k) = [x^k] Sum_{j=0..n-1} t(j, n) for n>=2 and 0<=k<=n-2 with t(j,n) = (-1)^(j+n)*binomial(n,j)*(1-(1-x)^(n-1-j))*x^(-1)*n^j.
T(n,k) = [x^k] ((-n-x+1)^n+(x-1)*(1-n)^n-(-n)^n*x)*(-1)^n/((x-1)*x). (End)

cp's OEIS Frontend

A345632 Sum of terms of even index in the binomial decomposition of n^(n-1).

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A345633 Sum of terms of odd index in the binomial decomposition of n^(n-1).

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A346963 Decimal expansion of Integral_{x=-1/e..0} LambertW(x)*LambertW(-1,x) dx.

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A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

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A037184 Functional digraphs with 1 node not in the image.

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A090650 n^(n+6).

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A122606 n^(n+1) mod 7.

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A134574 Array, a(n,k) = total size of all n-length words on a k-letter alphabet, read by antidiagonals.

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A173249 Partial sums of A000272.

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A281620 Triangle read by rows: Poincaré polynomials of orbifold of Fermat hypersurfaces.