A104872 -id:A104872 - cp's OEIS frontend

A004248 Array read by ascending antidiagonals: A(n, k) = k^n.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 16, 27, 16, 5, 1, 0, 1, 32, 81, 64, 25, 6, 1, 0, 1, 64, 243, 256, 125, 36, 7, 1, 0, 1, 128, 729, 1024, 625, 216, 49, 8, 1, 0, 1, 256, 2187, 4096, 3125, 1296, 343, 64, 9, 1, 0, 1, 512, 6561, 16384, 15625, 7776, 2401, 512, 81, 10, 1

Offset: 0

David W. Wilson

This array transforms into A371761 using the Akiyama-Tanigawa algorithm for powers applied to the rows. - Peter Luschny, Apr 16 2024

This array transforms into A344499 using the Akiyama-Tanigawa algorithm for powers applied to the columns. - Peter Luschny, Apr 27 2024

			Seen as an array that is read by ascending antidiagonals:
[0] 1, 1,   1,    1,     1,     1,      1,      1,       1, ...
[1] 0, 1,   2,    3,     4,     5,      6,      7,       8, ...
[2] 0, 1,   4,    9,    16,    25,     36,     49,      64, ...
[3] 0, 1,   8,   27,    64,   125,    216,    343,     512, ...
[4] 0, 1,  16,   81,   256,   625,   1296,   2401,    4096, ...
[5] 0, 1,  32,  243,  1024,  3125,   7776,  16807,   32768, ...
[6] 0, 1,  64,  729,  4096, 15625,  46656, 117649,  262144, ...
[7] 0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, ...

T. D. Noe, Rows n=0..50 of triangle, flattened
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015.

For other versions see A051129 and A009998.
Row sums are A026898, diagonal sums are A104872. [Paul Barry, Mar 28 2005]
Cf. A003992, A048723, A371761, A344499.

T[x_, y_] := If[y == 0, 1, (x - y)^y];
Table[T[x, y], {x, 0, 11}, {y, x, 0, -1}] // Flatten (* Jean-François Alcover, Dec 15 2017 *)

T(x, y) = x^y \\ Charles R Greathouse IV, Feb 07 2017

def Arow(n, len): return [k**n for k in range(len)]
for n in range(8): print([n], Arow(n, 9))  # Peter Luschny, Apr 16 2024

Table of x^y, where (x,y) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...
As a number triangle, columns have g.f. x^k/(1 - kx). - Paul Barry, Mar 28 2005
From Paul Barry, Jul 13 2005: (Start)
T(n, k) = if(k <= n, k^(n - k), 0).
T(n, k) = Sum_{j=0..floor((n-k)/2)} (-1)^j*C(n-k, j)*C(n-k-j, n-k)*k^(n-k-2j).
(End)

New name by Peter Luschny, Apr 16 2024.

A352981 a(n) = Sum_{k=0..floor(n/2)} k^n.

Original entry on oeis.org

1, 0, 1, 1, 17, 33, 794, 2316, 72354, 282340, 10874275, 53201625, 2438235715, 14350108521, 762963987380, 5249352196144, 317685943157892, 2502137235710736, 169842891165484965, 1506994510201252425, 113394131858832552133, 1119223325228757961465

Offset: 0

Seiichi Manyama, Apr 13 2022

Seiichi Manyama, Table of n, a(n) for n = 0..429

Cf. A089072, A104872, A352982, A352983.

[(&+[k^n: k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Nov 01 2022

a[0] = 1; a[n_] := Sum[k^n, {k, 0, Floor[n/2]}]; Array[a, 22, 0] (* Amiram Eldar, Apr 13 2022 *)

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

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

[sum( k^n for k in range((n//2)+1)) for n in range(41)] # G. C. Greubel, Nov 01 2022

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

a(n) ~ exp((3 + (-1)^n)/2) * (n/2)^n / (exp(2) - 1). - Vaclav Kotesovec, Apr 14 2022

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

A345747 a(n) = n! * Sum_{k=0..floor(n/2)} k^(n - 2*k)/k!.

Original entry on oeis.org

1, 0, 2, 6, 36, 240, 2280, 27720, 425040, 7862400, 171188640, 4319330400, 125199708480, 4142318019840, 155388782989440, 6557345831836800, 308677784640825600, 16079233115648102400, 920518264903690252800, 57603377545940850624000

Offset: 0

Seiichi Manyama, Sep 17 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..315

Cf. A354436, A355575.

Cf. A104872, A216507, A356628.

Join[{1}, Table[n!*Sum[k^(n - 2*k)/k!, {k, 0, n/2}], {n, 1, 20}]] (* Vaclav Kotesovec, Oct 30 2022 *)

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

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

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

a(n) ~ sqrt(2*Pi) * exp((n - 1/2)/LambertW(exp(2/3)*(2*n - 1)/6) - 2*n) * n^(2*n + 1/2) / (3^(n + 1/2) * sqrt(1 + LambertW(exp(2/3)*(2*n - 1)/6)) * LambertW(exp(2/3)*(2*n - 1)/6)^n). - Vaclav Kotesovec, Oct 30 2022

A352945 a(n) = Sum_{k=0..floor(n/3)} k^(n-3*k).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 3, 5, 10, 20, 42, 93, 214, 516, 1307, 3473, 9659, 28002, 84257, 262229, 842196, 2787864, 9506796, 33388393, 120727844, 449148808, 1717595949, 6743420017, 27147152525, 111931584098, 472225684599, 2037019695797, 8979468552886, 40432306870108

Offset: 0

Seiichi Manyama, Apr 09 2022

Seiichi Manyama, Table of n, a(n) for n = 0..738

Cf. A104872, A355575.

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

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^(3*k)/(1-k*x)))

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

a(n) ~ sqrt(2*Pi/3) * (n/(3*LambertW(exp(1)*n/3)))^(n + 1/2 - n/LambertW(exp(1)*n/3)) / sqrt(1 + LambertW(exp(1)*n/3)). - Vaclav Kotesovec, Apr 14 2022

A353288 a(n) = Sum_{k=0..floor(n/2)} k^(n-2k) Stirling2(n-k,k).

Original entry on oeis.org

1, 0, 1, 1, 2, 7, 30, 139, 723, 4487, 33551, 289854, 2774999, 29016343, 333139222, 4232908176, 59442337179, 912948755487, 15154215501815, 269933506466203, 5150440487875190, 105326085645729766, 2307425141636199329, 53998118146846356916, 1343998910355295080556

Offset: 0

Seiichi Manyama, Apr 09 2022

nonn

Cf. A119429, A171367, A353262.

Cf. A104872, A353287.

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

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

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

A352985 a(n) = Sum_{k=0..floor(n/2)} k^(2(n-2k)).

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 18, 74, 339, 1770, 10915, 79555, 663140, 6109351, 61264436, 669862580, 8044351557, 106331744724, 1536980041573, 24028469781765, 402558463751974, 7195932984364585, 137204787854813174, 2792969599543659326, 60668198155262809815

Offset: 0

Seiichi Manyama, Apr 13 2022

Cf. A104872, A352983.

a[0] = 1; a[n_] := Sum[k^(2*(n - 2*k)), {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 13 2022 *)

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

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

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

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

cp's OEIS Frontend

A004248 Array read by ascending antidiagonals: A(n, k) = k^n.

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A352981 a(n) = Sum_{k=0..floor(n/2)} k^n.

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

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A345747 a(n) = n! * Sum_{k=0..floor(n/2)} k^(n - 2*k)/k!.

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A352945 a(n) = Sum_{k=0..floor(n/3)} k^(n-3*k).

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A353288 a(n) = Sum_{k=0..floor(n/2)} k^(n-2*k) * Stirling2(n-k,k).

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A352985 a(n) = Sum_{k=0..floor(n/2)} k^(2*(n-2*k)).

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Formula

A353288 a(n) = Sum_{k=0..floor(n/2)} k^(n-2k) Stirling2(n-k,k).

A352985 a(n) = Sum_{k=0..floor(n/2)} k^(2(n-2k)).