A026898 -id:A026898 - cp's OEIS frontend

A038125 a(n) = Sum_{k=0..n} (k-n)^k.

1, 1, 0, 0, 1, -1, 0, 6, -19, 29, 48, -524, 2057, -3901, -9632, 129034, -664363, 1837905, 2388688, -67004696, 478198545, -1994889945, 1669470784, 56929813934, -615188040195, 3794477505573, -12028579019536, -50780206473220

Offset: 0

Jim Ferry (jferry(AT)alum.mit.edu)

			0^0 = 1,
1^0 - 0^1 = 1,
2^0 - 1^1 + 0^2 = 0,
3^0 - 2^1 + 1^2 - 0^3 = 0,
...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A003101, A026898, A120485.

Prepend[ Table[ Sum[ (k-n)^k, {k, 0, n} ], {n, 30} ], 1 ]

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1+k*x))) \\ Seiichi Manyama, Dec 02 2021

a(n) = sum(k=0, n, (k-n)^k); \\ Michel Marcus, Dec 03 2021

G.f.: 1+ sum(k>=0, x^(k+1)/(1+x^(k+1)) ) = 1/Q(0), where Q(k) = 1 - x + x^2*(k+1)/(1 + (k+1)*x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 10 2014

A104872 Diagonal sums of A004248.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 12, 27, 64, 163, 441, 1268, 3855, 12344, 41464, 145653, 533736, 2036149, 8071785, 33192790, 141351715, 622384730, 2829417276, 13263528351, 64038928728, 318121600695, 1624347614737, 8517247764136, 45822087138879

Offset: 0

Paul Barry, Mar 28 2005

Seiichi Manyama, Table of n, a(n) for n = 0..681

Cf. A004248, A026898, A352944.

a(n) = sum(k=0, n\2, k^(n-2*k)); \\ Seiichi Manyama, Apr 09 2022

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^(2*k)/(1-k*x))) \\ Seiichi Manyama, Apr 09 2022

a(n) = Sum_{k=0..floor(n/2)} k^(n-2*k).
G.f.: Sum_{k>=0} x^(2*k) / (1 - k * x). - Seiichi Manyama, Apr 09 2022
a(n) ~ sqrt(Pi) * (n/(2*LambertW(exp(1)*n/2)))^(n + 1/2 - n/LambertW(exp(1)*n/2)) / sqrt(1 + LambertW(exp(1)*n/2)). - Vaclav Kotesovec, Apr 14 2022

A234568 Sum_{k=0..n} (n-k)^(2*k).

Original entry on oeis.org

1, 1, 2, 6, 27, 163, 1268, 12344, 145653, 2036149, 33192790, 622384730, 13263528351, 318121600695, 8517247764136, 252725694989612, 8258153081400857, 295515712276222953, 11523986940937975402, 487562536078882116718, 22291094729329088403299, 1097336766599161926448779

Offset: 0

Paul D. Hanna, Dec 28 2013

nonn

			O.g.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 163*x^5 + 1268*x^6 +...
O.g.f.: A(x) = 1 + x/(1-x) + x^2/(1-4*x) + x^3/(1-9*x) + x^4/(1-16*x) +...
E.g.f.: E(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 27*x^4/4! + 163*x^5/5! +...
where the e.g.f. is a series involving iterated integration:
E(x) = 1 + Integral exp(x) dx + Integral^2 exp(4*x) dx^2 + Integral^3 exp(9*x) dx^3 + Integral^4 exp(16*x) dx^4 +...

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

Cf. A003101, A026898, A349880, A349881.

Flatten[{1,Table[Sum[(n-k)^(2*k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 23 2014 *)

a(n)=sum(k=0, n, (n-k)^(2*k))
for(n=0, 20, print1(a(n), ", "))

/* From o.g.f. Sum_{n>=0} x^n/(1-n^2*x): */
{a(n)=polcoeff(sum(m=0, n, x^m/(1-m^2*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

/* From e.g.f. involving iterated integration: */
INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G
a(n)=my(A=1+x);A=1+sum(k=1,n,INTEGRATE(k,exp(k^2*x+x*O(x^n))));n!*polcoeff(A,n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Dec 28 2013

O.g.f.: Sum_{n>=0} x^n / (1 - n^2*x).
E.g.f.: Sum_{n>=0} Integral^n exp(n^2*x) dx^n, where integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration.
a(n) ~ sqrt(Pi) * (n/LambertW(exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(exp(1)*n)) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A105795 Shallow diagonal sums of the triangle k!*Stirling2(n,k): a(n) = Sum_{k=0..floor(n/2)} T(n-k,k) where T is A019538.

Original entry on oeis.org

1, 0, 1, 1, 3, 7, 21, 67, 237, 907, 3741, 16507, 77517, 385627, 2024301, 11174587, 64673997, 391392667, 2470864941, 16237279867, 110858862477, 784987938907, 5755734591981, 43636725010747, 341615028340557, 2758165832945947, 22940755633301421, 196354180631212027

Offset: 0

Paul Barry, Apr 20 2005

From Gus Wiseman, Jan 08 2019: (Start)
Also the number of set partitions of {1,...,n} into blocks of sizes > 1 whose minima form an initial interval of positive integers. For example, the a(5) = 7 set partitions are:
  {{1,2,3,4,5}}
  {{1,3},{2,4,5}}
  {{1,4},{2,3,5}}
  {{1,5},{2,3,4}}
  {{1,3,4},{2,5}}
  {{1,3,5},{2,4}}
  {{1,4,5},{2,3}}
Also the number of ordered set partitions of {1,...,n-k} of length k, for any 0 <= k <= n. For example, the a(5) = 7 ordered set partitions are:
  {{1,2,3,4}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1,2},{3}}
  {{1,3},{2}}
  {{2,3},{1}}
(End)

			a(8) = 1!*Stirling2(7,1) + 2!*Stirling2(6,2) + 3!*Stirling2(5,3) + 4!*Stirling2(4,4) = 1 + 62 + 150 + 24 = 237. - _Peter Bala_, Jul 09 2014

Alois P. Heinz, Table of n, a(n) for n = 0..599
Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024. See p. 13.

Cf. A000110, A000296, A000670, A003101, A008277, A019538, A026898, A287215.

a:= n-> add(Stirling2(n-k, k)*k!, k=0..n/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 09 2014

Table[Sum[StirlingS2[n-k, k]*k!, {k,0,n/2}],{n,0,20}] (* Vaclav Kotesovec, Jul 16 2014 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Length[Join@@Permutations/@Select[sps[Range[n-k]],Length[#]==k&]],{k,0,n}],{n,0,10}] (* Gus Wiseman, Jan 08 2019 *)

/* From Paul Barry's formula: */
{a(n)=sum(k=0,floor(n/2), sum(i=0,k,(-1)^i*binomial(k, i)*(k-i)^(n-k)))} \\ Paul D. Hanna, Dec 28 2013
for(n=0,30,print1(a(n),", "))

/* From e.g.f. series involving iterated integration: */
{INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}
{a(n)=local(A=1+x);A=1+sum(k=1,n,INTEGRATE(k,(exp(x+x*O(x^n))-1)^k ));n!*polcoeff(A,n)} \\ Paul D. Hanna, Dec 28 2013

a(n) = sum{k=0..floor(n/2), sum{(-1)^i*binomial(k, i)*(k-i)^(n-k)}}.
E.g.f.: Sum_{n>=0} Integral^n (exp(x) - 1)^n dx^n, where integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration. - Paul D. Hanna, Dec 28 2013
Formal o.g.f.: 1/(1 + x)*sum {n >= 0} 1/(1 - n*x)*(x/(1 + x))^n = 1 + x^2 + x^3 + 3*x^4 + 7*x^5 + .... Cf. A229046. - Peter Bala, Jul 09 2014

A349880 Expansion of Sum_{k>=0} x^k/(1 - k^3 * x).

Original entry on oeis.org

1, 1, 2, 10, 93, 1307, 28002, 842196, 33388393, 1717595949, 111931584098, 8979468552886, 872315432217509, 101425775048588759, 13924209725224120770, 2229705716369149960592, 412760812611799202662609, 87644186710319273062637625, 21180850892383599137766296770

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..246

Cf. A026898, A234568, A349881.

Cf. A349857.

a(n, s=0, t=3) = sum(k=0, n, k^(t*(n-k)+s));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^3*x)))

a(n) = Sum_{k=0..n} k^(3*(n-k)).

a(n) ~ sqrt(2*Pi/3) * (n/LambertW(exp(1)*n))^(1/2 + 3*n - 3*n/LambertW(exp(1)*n)) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A062810 a(n) = Sum_{i=1..n} i^(n - i) + (n - i)^i.

Original entry on oeis.org

1, 3, 7, 17, 45, 131, 419, 1465, 5561, 22755, 99727, 465537, 2303829, 12037571, 66174411, 381560425, 2301307841, 14483421859, 94909491607, 646309392369, 4565559980989, 33401808977411, 252713264780595, 1974606909857945

Offset: 1

Olivier Gérard, Jun 23 2001

Seiichi Manyama, Table of n, a(n) for n = 1..600

Cf. A003101, A026898, A062811, A062812.

Mathematica
```
Sum[i^(n - i) + (n - i)^i, {i, 1, n}]
```

a(n) = sum(i=1, n, i^(n-i) + (n-i)^i); \\ Michel Marcus, Mar 24 2019

a(n) = 2 * A026898(n-1) - 1.

a(n) = 2 * A003101(n-1) + 1.

A103439 a(n) = Sum_{i=0..n-1} Sum_{j=0..i} (i-j+1)^j.

Original entry on oeis.org

0, 1, 3, 7, 16, 39, 105, 315, 1048, 3829, 15207, 65071, 297840, 1449755, 7468541, 40555747, 231335960, 1381989881, 8623700811, 56078446615, 379233142800, 2662013133295, 19362917622001, 145719550012299, 1133023004941272, 9090156910550109, 75161929739797519

Offset: 0

Ralf Stephan, Feb 11 2005

nonn

Partial sums of A026898.
Antidiagonal sums of array A103438.
Row sums of A123490. - Paul Barry, Oct 01 2006

Alois P. Heinz, Table of n, a(n) for n = 0..600
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.

Cf. A026898, A103438, A123490.

[0] cat [(&+[ (&+[ (k-j+1)^j : j in [0..k]]) : k in [0..n-1]]): n in [1..30]]; // G. C. Greubel, Jun 15 2021

b:= proc(i) option remember; add((i-j+1)^j, j=0..i) end:
a:= proc(n) option remember; add(b(i), i=0..n-1) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 02 2019

Join[{0},Table[Sum[Sum[(i-j+1)^j,{j,0,i}],{i,0,n}],{n,0,30}]] (* Harvey P. Dale, Dec 03 2018 *)

a(n) = sum(i=0, n-1, sum(j=0, i, (i-j+1)^j)); \\ Michel Marcus, Jun 15 2021

[sum(sum((k-j+1)^j for j in (0..k)) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, Jun 15 2021

a(n+1) = Sum_{k=0..n} ((k+2)^(n-k) + k)/(k+1). - Paul Barry, Oct 01 2006

G.f.: (G(0)-1)/(1-x) where G(k) = 1 + x*(2*k*x-1)/(2*k*x+x-1 - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013

Name edited by Alois P. Heinz, Dec 02 2019

A352944 a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^k.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 31, 61, 125, 266, 579, 1305, 3009, 7120, 17255, 42697, 108005, 278466, 731883, 1958589, 5331625, 14758720, 41501135, 118507301, 343405709, 1009313322, 3007557523, 9081204849, 27775308049, 86014412384, 269603741111, 855012176081

Offset: 0

Seiichi Manyama, Apr 09 2022

Cf. A026898, A104872, A352946.

Cf. A087811.

Join[{1},Table[Sum[(n-2k)^k,{k,0,Floor[n/2]}],{n,40}]] (* Harvey P. Dale, Dec 12 2022 *)

PARI
```
a(n) = sum(k=0, n\2, (n-2*k)^k);
```

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k*x^2)))

G.f.: Sum_{k>=0} x^k / (1 - k * x^2).

a(n) ~ sqrt(Pi) * (n/LambertW(exp(1)*n))^((n + 1 - n/LambertW(exp(1)*n))/2) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Apr 14 2022

A104879 Row sums of a sum-of-powers triangle.

Original entry on oeis.org

1, 2, 4, 8, 17, 40, 106, 316, 1049, 3830, 15208, 65072, 297841, 1449756, 7468542, 40555748, 231335961, 1381989882, 8623700812, 56078446616, 379233142801, 2662013133296, 19362917622002, 145719550012300, 1133023004941273, 9090156910550110, 75161929739797520

Offset: 0

Paul Barry, Mar 28 2005

Row sums of A104878.

Cf. A103439 (terms differ by 1), A026898 (first differences).

a(n) = 1 + n + Sum_{k=2..n+1} (k^(n-k+1)-1)/(k-1).

a(n) = 1 + A103439(n). - Mathew Englander, Dec 19 2020

A327712 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all compositions of n into distinct parts (k is a composition length).

Original entry on oeis.org

1, 1, 1, 3, 3, 9, 29, 57, 135, 615, 2635, 6273, 25151, 82623, 525281, 2941047, 9100709, 38766777, 205155713, 902705793, 7714938567, 52987356783, 204844103977, 1042657233471, 5520661314689, 38159472253821, 211945677298567, 2404720648663335, 19773733727088813

Offset: 0

Alois P. Heinz, Sep 22 2019

nonn

Number of partitions of [n] with distinct block sizes such that each block contains exactly one block size as an element. a(5) = 9: 12345, 1235|4, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 1|2345.

Alois P. Heinz, Table of n, a(n) for n = 0..706
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
Wikipedia, Partition of a set

Cf. A026898, A326493, A327711, A364281.

with(combinat):
a:= n-> add(multinomial(n-nops(p), map(x-> x-1, p)[], 0), p=map(h->
    permute(h)[], select(l-> nops(l)=nops({l[]}), partition(n)))):
seq(a(n), n=0..28);
# second Maple program:
a:= proc(m) option remember; local b; b:=
      proc(n, i, j) option remember; `if`(i*(i+1)/2>=n,
       `if`(n=0, (m-j)!*j!, b(n, i-1, j)+
        b(n-i, min(n-i, i-1), j+1)/(i-1)!), 0)
      end: b(m$2, 0):
    end:
seq(a(n), n=0..28);

a[m_] := a[m] = Module[{b}, b[n_, i_, j_] := b[n, i, j] = If[i(i + 1)/2 >= n, If[n == 0, (m - j)! j!, b[n, i - 1, j] + b[n - i, Min[n - i, i - 1], j + 1]/(i - 1)!], 0]; b[m, m, 0]];
a /@ Range[0, 28] (* Jean-François Alcover, May 10 2020, after 2nd Maple program *)

cp's OEIS Frontend

A038125 a(n) = Sum_{k=0..n} (k-n)^k.

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A104872 Diagonal sums of A004248.

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A234568 Sum_{k=0..n} (n-k)^(2*k).

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A105795 Shallow diagonal sums of the triangle k!*Stirling2(n,k): a(n) = Sum_{k=0..floor(n/2)} T(n-k,k) where T is A019538.

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A349880 Expansion of Sum_{k>=0} x^k/(1 - k^3 * x).

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A062810 a(n) = Sum_{i=1..n} i^(n - i) + (n - i)^i.

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A103439 a(n) = Sum_{i=0..n-1} Sum_{j=0..i} (i-j+1)^j.

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