A120485 -id:A120485 - cp's OEIS frontend

A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.

1, 1, 4, 1, 8, 27, 1, 16, 81, 256, 1, 32, 243, 1024, 3125, 1, 64, 729, 4096, 15625, 46656, 1, 128, 2187, 16384, 78125, 279936, 823543, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489

Offset: 1

Alford Arnold, Dec 04 2003

T(n, k) = number of mappings from an n-element set into a k-element set. - Clark Kimberling, Nov 26 2004
Let S be the semigroup of (full) transformations on [n].  Let a be in S with rank(a) = k.  Then T(n,k) = |a S|, the number of elements in the right principal ideal generated by a. - Geoffrey Critzer, Dec 30 2021
From Manfred Boergens, Jun 23 2024: (Start)
In the following two comments the restriction k<=n can be lifted, allowing all k>=1.
T(n,k) is the number of n X k binary matrices with row sums = 1.
T(n,k) is the number of coverings of [n] by tuples (A_1,...,A_k) in P([n])^k with disjoint A_j, with P(.) denoting the power set.
For nonempty A_j see A019538.
For tuples with "disjoint" dropped see A092477.
For tuples with nonempty A_j and with "disjoint" dropped see A218695. (End)

			Triangle begins:
  1;
  1,  4;
  1,  8,  27;
  1, 16,  81,  256;
  1, 32, 243, 1024,  3125;
  1, 64, 729, 4096, 15625, 46656;
  ...

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.1(ii).

Related to triangle of Eulerian numbers A008292.
Cf. A000312, A007778, A008788, A031971 (row sums), A062206, A083282.
Cf. A085524, A085526, A120485, A226065, A352981, A252982.
Cf. A019538, A092477, A218695.

a089072 = flip (^)
a089072_row n = map (a089072 n) [1..n]
a089072_tabl = map a089072_row [1..]  -- Reinhard Zumkeller, Mar 18 2013

[k^n: k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 01 2022

Column[Table[k^n, {n, 8}, {k, n}], Center] (* Alonso del Arte, Nov 14 2011 *)

flatten([[k^n for k in range(1,n+1)] for n in range(1,12)]) # G. C. Greubel, Nov 01 2022

Sum_{k=1..n} T(n, k) = A031971(n).
T(n, n) = A000312(n).
T(2*n, n) = A062206(n).
a(n) = (n + T*(1-T)/2)^T, where T = round(sqrt(2*n),0). - Gerald Hillier, Apr 12 2015
T(n,k) = A051129(n,k). - R. J. Mathar, Dec 10 2015
T(n,k) = Sum_{i=0..k} Stirling2(n,i)*binomial(k,i)*i!. - Geoffrey Critzer, Dec 30 2021
From G. C. Greubel, Nov 01 2022: (Start)
T(n, n-1) = A007778(n-1), n >= 2.
T(n, n-2) = A008788(n-2), n >= 3.
T(2*n+1, n) = A085526(n).
T(2*n-1, n) = A085524(n).
T(2*n-1, n-1) = A085526(n-1), n >= 2.
T(3*n, n) = A083282(n).
Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A120485(n).
Sum_{k=1..floor(n/2)} T(n-k, k) = A226065(n).
Sum_{k=1..floor(n/2)} T(n, k) = A352981(n).
Sum_{k=1..floor(n/3)} T(n, k) = A352982(n). (End)

More terms and better definition from Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

Offset corrected by Reinhard Zumkeller, Mar 18 2013

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

Original entry on oeis.org

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

A349884 Expansion of Sum_{k>=0} (k * x)^k/(1 + k^2 * x).

Original entry on oeis.org

1, 1, 3, 12, 76, 961, 15407, 221528, 3260936, 80774113, 2462081967, 50963779604, 922244742292, 61063845514113, 2868669700179871, 2019727494212912, -47889136910252848, 461395118866593115713, 5781219348638565771423, -2108738296748190078596084

Offset: 0

Seiichi Manyama, Dec 03 2021

sign

Cf. A120485, A349885.

Cf. A349856, A349859, A349863.

a[n_] := Sum[If[k == 2*n - k == 0, 1, (-1)^(n - k) * k^(2*n - k)], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 04 2021 *)

a(n, t=2) = sum(k=0, n, (-k^t)^(n-k)*k^k);

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1+k^2*x)))

a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n-k).

A349889 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n).

Original entry on oeis.org

1, 1, 15, 666, 59230, 8775075, 1948891581, 605698755508, 250914820143996, 133610836793706405, 88919025666286620475, 72317513878698256697166, 70571883548815735717843290, 81383769918571603591381635271

Offset: 0

Seiichi Manyama, Dec 04 2021

nonn

Cf. A120485, A249459, A349884, A349891, A349902.

Join[{1},Table[Sum[(-1)^(n-k) k^(2n),{k,0,n}],{n,20}]] (* Harvey P. Dale, Nov 19 2023 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k^(2*n));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1+k^2*x)))

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

a(n) ~ 1/(1 + exp(-2)) * n^(2*n). - Vaclav Kotesovec, Dec 10 2021

A349891 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(k*n).

Original entry on oeis.org

1, 0, 16, 19619, 4294436111, 298022124379673232, 10314423867168242405282727694, 256923577039829077600620024397823949901879, 6277101735175093150055816289268196664555481440709896684157

Offset: 0

Seiichi Manyama, Dec 04 2021

nonn

Cf. A120485, A349886, A349889.

a(n) = sum(k=0, n, (-1)^(n-k)*k^(k*n));

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, k^k^2*x^k/(1+k^k*x)))

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

a(n) ~ n^(n^2). - Vaclav Kotesovec, Dec 10 2021

A091884 Triangle of numbers defined by Knuth.

Original entry on oeis.org

1, 1, 1, 4, 3, 3, 27, 19, 20, 20, 256, 175, 191, 190, 190, 3125, 2101, 2344, 2312, 2313, 2313, 46656, 31031, 35127, 34398, 34462, 34461, 34461, 823543, 543607, 621732, 605348, 607535, 607407, 607408, 607408, 16777216, 11012415, 12692031, 12301406, 12366942, 12360381, 12360637, 12360636, 12360636

Offset: 0

Michael Somos, Feb 08 2004

			Triangle begins:
     1;
     1,    1;
     4,    3,    3;
    27,   19,   20,   20;
   256,  175,  191,  190,  190;
  3125, 2101, 2344, 2312, 2313, 2313;
  ...

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 6.4 Answer to Exer. 46.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.

Seiichi Manyama, Rows n = 0..139, flattened

Column k=0..1 give A000312, A045531.

Main diagonal gives A120485.

T(n,k)=if(k<0 || k>n,0,sum(j=0,k,(-1)^j*(n-j)^n))

T(n,k) = Sum_{j=0..k} (-1)^j * (n-j)^n.

A332627 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * k^n.

Original entry on oeis.org

1, 1, 6, 117, 4388, 266065, 23731314, 2923345621, 475364541672, 98623225721601, 25421365316232710, 7969388199705535141, 2985785305877403047820, 1317500933136749853197329, 676266417871227455138941242, 399516621958550611386236160405

Offset: 0

Ilya Gutkovskiy, Apr 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A000166, A120485, A134095, A302397, A319392, A332408.

Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(-x))^k))) \\ Seiichi Manyama, Feb 19 2022

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 + k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A072364 = exp(-exp(-1)). - Vaclav Kotesovec, Jul 10 2021
E.g.f.: Sum_{k>=0} (k*x*exp(-x))^k. - Seiichi Manyama, Feb 19 2022

A349885 Expansion of Sum_{k>=0} (k * x)^k/(1 + k^3 * x).

Original entry on oeis.org

1, 1, 3, -4, -218, 4377, 189549, -13317056, -283835940, 117015022505, -5604964950389, -1791024716075124, 422751913131376674, 8850160172208790801, -30082452518043880807911, 7173002090013176579439392, 1556433498641034120823054072

Offset: 0

Seiichi Manyama, Dec 03 2021

sign

Cf. A120485, A349884.

a[n_] := Sum[If[k == 3*n - 2*k == 0, 1, (-1)^(n - k) * k^(3*n - 2*k)], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Dec 04 2021 *)

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

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(3*n-2*k).

A349902 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(3*n).

Original entry on oeis.org

1, 1, 63, 19172, 16249870, 29458152441, 97813591721181, 537081363012854224, 4535464309375188976956, 55796581668379082029481225, 958824462567528346234944706075, 22255431432328421226838750870120356, 678866987929798923743810982299237129610

Offset: 0

Seiichi Manyama, Dec 05 2021

nonn

Cf. A120485, A349885, A349889, A349891.

Join[{1},Table[Sum[(-1)^(n-k) k^(3n),{k,0,n}],{n,20}]] (* Harvey P. Dale, Apr 12 2022 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k^(3*n));

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

G.f.: Sum_{k>=0} (k^3 * x)^k/(1 + k^3 * x).

a(n) ~ 1/(1 + exp(-3)) * n^(3*n). - Vaclav Kotesovec, Dec 10 2021

A368466 a(n) = Sum_{k=0..n} 2^k * k^n.

Original entry on oeis.org

1, 2, 18, 250, 4810, 118458, 3557610, 126109562, 5153959338, 238596116794, 12340467941098, 705262375055610, 44135963944338474, 3001795007526424250, 220466095716711140202, 17389850740043552754298, 1466156761178169939270826, 131580021359494993268692026

Offset: 0

Seiichi Manyama, Dec 25 2023

Main diagonal of A368479.

Cf. A031971, A120485, A349963.

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

a(n) ~ 2^n * n^n / (1 - exp(-1)/2). - Vaclav Kotesovec, Dec 26 2023

cp's OEIS Frontend

A089072 Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.

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

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A349884 Expansion of Sum_{k>=0} (k * x)^k/(1 + k^2 * x).

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A349889 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n).

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A349891 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(k*n).

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A091884 Triangle of numbers defined by Knuth.

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A332627 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * k^n.

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A349885 Expansion of Sum_{k>=0} (k * x)^k/(1 + k^3 * x).