A062024 -id:A062024 - cp's OEIS frontend

A302583 a(n) = ((n + 1)^n - (n - 1)^n)/2.

0, 1, 4, 28, 272, 3376, 51012, 908608, 18640960, 432891136, 11225320100, 321504185344, 10079828372880, 343360783937536, 12627774819845668, 498676704524517376, 21046391759976988928, 945381827279671853056, 45032132922921758270916, 2267322327322331161821184

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000169, A065440, A007778, A062024, A115416, A274278, A293022, A302584, A302585, A302586, A302587.

Table[((n + 1)^n - (n - 1)^n)/2, {n, 0, 19}]
nmax = 19; CoefficientList[Series[(x^2 - LambertW[-x]^2)/(2 x LambertW[-x] (1 + LambertW[-x])), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! SeriesCoefficient[Exp[n x] Sinh[x], {x, 0, n}], {n, 0, 19}]

E.g.f.: (x^2 - LambertW(-x)^2)/(2*x*LambertW(-x)*(1 + LambertW(-x))).

a(n) = n! * [x^n] exp(n*x)*sinh(x).

A302587 a(n) = n! * [x^n] exp(nx)tanh(x).

Original entry on oeis.org

0, 1, 4, 25, 224, 2641, 38592, 671665, 13548544, 310580161, 7971353600, 226406902921, 7049219383296, 238722074157841, 8735529994928128, 343474252543881313, 14441163232204292096, 646510839624706118401, 30704150325602206089216, 1541807339347429264648441

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Cf. A000182, A009832, A062024, A302583, A302584, A302585, A302586.

N:= 100: # to get a(0)..a(N)
T:= series(tanh(x),x,N+1):
C:= [seq(coeff(T,x,j),j=1..N)]:
seq(n! * add(C[i]*n^(n-i)/(n-i)!,i=1..n,2), n=0..N); # Robert Israel, Apr 10 2018

Table[n! SeriesCoefficient[Exp[n x] Tanh[x], {x, 0, n}], {n, 0, 19}]

a(n) = my(x='x+O('x^(n+1))); polcoeff(n!*exp(n*x)*tanh(x), n); \\ Michel Marcus, Apr 11 2018; corrected Jun 15 2022

a(n) ~ tanh(1) * n^n. - Vaclav Kotesovec, Jun 08 2019

A084061 Square number array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 5, 27, 1, 1, 6, 36, 256, 1, 1, 7, 45, 353, 3125, 1, 1, 8, 54, 452, 4400, 46656, 1, 1, 9, 63, 553, 5725, 66637, 823543, 1, 1, 10, 72, 656, 7100, 87704, 1188544, 16777216, 1, 1, 11, 81, 761, 8525, 109863, 1577849, 24405761, 387420489, 1, 1, 12, 90, 868, 10000, 133120, 1991752, 32618512, 567108864, 10000000000

Offset: 0

Paul Barry, May 11 2003

			Rows begin:
1 1 4 27 256 ...
1 1 5 36 353 ...
1 1 6 45 452 ...
1 1 7 54 553 ...
1 1 8 63 656 ...

G. C. Greubel, Antidiagonal rows n = 0..100, flattened

Rows include A000312, A062024, A084063, A084064, A084065.

Diagonals include A084062, A084063, A084095.

Flat(List([0..10], n-> List([0..n], k-> ((k+Sqrt(n-k))^k + (k-Sqrt(n-k))^k)/2 ))); # G. C. Greubel, Jan 11 2020

[Round(((k+Sqrt(n-k))^k + (k-Sqrt(n-k))^k)/2): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 11 2020

seq(seq( round(((k+sqrt(n-k))^k + (k-sqrt(n-k))^k)/2), k=0..n), n=0..10); # G. C. Greubel, Jan 11 2020

Table[If[n==0 && k==0, 1, Round[((k-Sqrt[n-k])^k + (k+Sqrt[n-k])^k)/2]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 11 2020 *)

T(n,k) = round( ((k+sqrt(n-k))^n + (k-sqrt(n-k))^k)/2 ); \\ G. C. Greubel, Jan 11 2020

[[round(((k+sqrt(n-k))^k + (k-sqrt(n-k))^k)/2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 11 2020

T(n, k) = ( (n - sqrt(k))^n + (n + sqrt(k))^n )/2.

A302584 a(n) = n! * [x^n] exp(n*x)/cos(x).

Original entry on oeis.org

1, 1, 5, 36, 357, 4500, 68857, 1239504, 25661545, 600655824, 15684383021, 452001644864, 14249852124365, 487836995500608, 18022519535240417, 714658089577017600, 30275849571771536977, 1364687729891761740032, 65213822241378992547925, 3293203845745202062590976

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000364, A003701, A062024, A115415, A115416, A302583, A302585, A302586, A302587.

Table[n! SeriesCoefficient[Exp[n x]/Cos[x], {x, 0, n}], {n, 0, 19}]
Table[(2 I)^n EulerE[n, (1 - I n)/2], {n, 0, 19}]

a(n) ~ n^n / cos(1). - Vaclav Kotesovec, Jun 08 2019

A302585 a(n) = n! * [x^n] exp(n*x)/cosh(x).

Original entry on oeis.org

1, 1, 3, 18, 165, 2000, 29855, 527632, 10762857, 248811264, 6428081979, 183537694208, 5739195739277, 195059957567488, 7159662639822615, 282252719348582400, 11894243092571825745, 533554809104057434112, 25384473065818477067123, 1276688324194885747474432

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000364, A062024, A119880, A119881, A155585, A302583, A302584, A302586, A302587.

Table[n! SeriesCoefficient[Exp[n x]/Cosh[x], {x, 0, n}], {n, 0, 19}]
Table[2^n EulerE[n, (n + 1)/2], {n, 0, 19}]

a(n) ~ n^n / cosh(1). - Vaclav Kotesovec, Jun 08 2019

A084064 Third row of number array A084061.

Original entry on oeis.org

1, 1, 6, 45, 452, 5725, 87704, 1577849, 32618512, 762046137, 19856872032, 571007744549, 17962793210944, 613650073693397, 22624291883495808, 895379458590349425, 37861032312753094912, 1703550488551604490353

Offset: 0

Paul Barry, May 11 2003

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A000312, A062024, A084061, A084062, A084063, A084065, A084095, A084096.

List([0..20], n-> ((n-Sqrt(2))^n + (n+Sqrt(2))^n)/2); # G. C. Greubel, Jan 11 2020

[Round(((n-Sqrt(2))^n + (n+Sqrt(2))^n)/2): n in [0..20]]; // G. C. Greubel, Jan 11 2020

seq( round(((n-sqrt(2))^n + (n+sqrt(2))^n)/2), n=0..20); # G. C. Greubel, Jan 11 2020

Table[Round[((n+Sqrt[2])^n + (n-Sqrt[2])^n)/2], {n,0,20}] (* G. C. Greubel, Jan 11 2020 *)

vector(21, n, round(((n-1-sqrt(2))^(n-1) + (n-1+sqrt(2))^(n-1))/2) ) \\ G. C. Greubel, Jan 11 2020

[round(((n-sqrt(2))^n + (n+sqrt(2))^n)/2) for n in (0..20)] # G. C. Greubel, Jan 11 2020

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

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

Original entry on oeis.org

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.

A262718 a(n) = (n+1)^n - 2*(n^n) + (n-1)^n.

Original entry on oeis.org

0, 0, 2, 18, 194, 2550, 39962, 730002, 15257090, 359376750, 9424209002, 272385029466, 8604312602690, 294957765448710, 10906288759973882, 432701819402940450, 18336112083960655874, 826578941145375829470, 39497618599385891373002, 1994276034034710498109674

Offset: 0

Vladimir Kruchinin, Sep 28 2015

Obviously, a(n) is always an even number. - Altug Alkan, Sep 28 2015

Cf. A000169, A000312, A062024.

[(n+1)^n - 2*(n^n) + (n-1)^n: n in [0..30]]; // Vincenzo Librandi, Sep 28 2015

Join[{0}, Table[(n + 1)^n - 2 (n^n) + (n - 1)^n, {n, 30}]] (* Vincenzo Librandi, Sep 28 2015 *)

B(x):=-lambert_w(-x);
makelist(n!*coeff(taylor(diff(B(x),x)*(1-x/B(x))^2,x,0,20),x,n),n,0,10);

a(n) = (n+1)^n - 2*(n^n) + (n-1)^n;
vector(30, n, a(n-1)) \\ Altug Alkan, Sep 28 2015

E.g.f.: A(x) = B'(x)*(1-x/B(x))^2, where B(x) is g.f. of A000169.
a(n) = Sum{k=1..n} (k!*binomial(n-1,k-2)*stirling2(n,k)), n>0, a(0)=0.
a(n) = 2*(A062024(n) - A000312(n)). - Michel Marcus, Sep 28 2015

cp's OEIS Frontend

A302583 a(n) = ((n + 1)^n - (n - 1)^n)/2.

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A302587 a(n) = n! * [x^n] exp(n*x)*tanh(x).

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Maple

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A084061 Square number array read by antidiagonals.

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A302584 a(n) = n! * [x^n] exp(n*x)/cos(x).

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A302585 a(n) = n! * [x^n] exp(n*x)/cosh(x).

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A084064 Third row of number array A084061.

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Magma

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Sage

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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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A302587 a(n) = n! * [x^n] exp(nx)tanh(x).