A121867 -id:A121867 - cp's OEIS frontend

A357728 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * (exp(x) - 1) ).

1, 1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, -2, -3, 0, 1, 0, -3, -6, -6, 0, 1, 0, -4, -9, -10, -5, 0, 1, 0, -5, -12, -12, 10, 33, 0, 1, 0, -6, -15, -12, 45, 190, 266, 0, 1, 0, -7, -18, -10, 100, 465, 1106, 1309, 0, 1, 0, -8, -21, -6, 175, 852, 2394, 4438, 4905, 0, 1, 0, -9, -24, 0, 270, 1345, 4004, 7827, 9978, 11516, 0

Offset: 0

Seiichi Manyama, Oct 11 2022

			Square array begins:
  1,  1,   1,   1,   1,   1, ...
  0,  0,   0,   0,   0,   0, ...
  0, -1,  -2,  -3,  -4,  -5, ...
  0, -3,  -6,  -9, -12, -15, ...
  0, -6, -10, -12, -12, -10, ...
  0, -5,  10,  45, 100, 175, ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Bell Polynomial.

Columns k=0-4 give: A000007, A121867, A357725, A357726, A357727.
Main diagonal gives A357729.
Cf. A357681, A357720.

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

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
T(n, k) = round((Bell_poly(n, sqrt(k)*I)+Bell_poly(n, -sqrt(k)*I)))/2;

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

T(n,k) = ( Bell_n(sqrt(k) * i) + Bell_n(-sqrt(k) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

A357725 Expansion of e.g.f. cos( sqrt(2) * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, -2, -6, -10, 10, 190, 1106, 4438, 9978, -35250, -666622, -5657370, -35308182, -155215970, -128513870, 7051468022, 105057922906, 1042016038254, 8053738122466, 44608555196294, 48639210067658, -3200193654245442, -60669816166988654, -769281697485061994

Offset: 0

Seiichi Manyama, Oct 10 2022

sign

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Bell Polynomial.

Column k=2 of A357728.

Cf. A121867, A264036, A357736.

my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*(exp(x)-1)))))

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

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(2)*I)+Bell_poly(n, -sqrt(2)*I)))/2;

a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling2(n,2*k).
a(n) = 1; a(n) = -2 * Sum_{k=0..n-1} binomial(n-1, k) * A357736(k).
a(n) = ( Bell_n(sqrt(2) * i) + Bell_n(-sqrt(2) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

A121869 Monthly Problem 10791, first expression.

Original entry on oeis.org

-1, 1, 0, -5, -15, 104, 1827, 7893, -207000, -5646249, -47897675, 1479282600, 74711288407, 1396956334921, -21032523700672, -2719998717430365, -104158663871982343, -715846242343471272, 189941380201812700699, 14820744271258596866013, 507768838531742620183176

Offset: 0

N. J. A. Sloane, Sep 05 2006

sign

G. C. Greubel, Table of n, a(n) for n = 0..330
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Cf. A121870, A024429, A024430, A121867, A121868.

List([0..25], n-> (-1)*Sum([0..n], k-> Stirling2(n,k)) *Sum([0..n], k-> (-1)^k*Stirling2(n,k)) ); # G. C. Greubel, Oct 08 2019

a:= func< n | (-1)*(&+[StirlingSecond(n,k): k in [0..n]])*(&+[ (-1)^k*StirlingSecond(n,k): k in [0..n]]) >;
[a(n): n in [0..25]]; // G. C. Greubel, Oct 08 2019

with(combinat): seq(-bell(n)*BellB(n, -1), n = 0..25); # G. C. Greubel, Oct 08 2019

Table[-BellB[n]*BellB[n, -1], {n,0,25}] (* G. C. Greubel, Oct 08 2019 *)

a(n) = (-1)*sum(k=0,n, stirling(n,k,2))*sum(k=0,n, (-1)^k*stirling(n,k,2));
vector(25, n, a(n-1)) \\ G. C. Greubel, Oct 08 2019

[ -sum(stirling_number2(n, k) for k in (0..n))*sum((-1)^k* stirling_number2(n,k) for k in (0..n)) for n in (0..25)] # G. C. Greubel, Oct 08 2019

a(n) = A024429(n)^2 - A024430(n)^2.

A121870 Monthly Problem 10791, second expression.

Original entry on oeis.org

1, 1, 2, 9, 61, 554, 6565, 96677, 1716730, 36072181, 881242577, 24674241834, 783024550969, 27896201305769, 1106485798248706, 48517267642373105, 2337333266369553253, 123040664089658462650, 7043260281573138384701, 436533086101058798529933

Offset: 0

N. J. A. Sloane, Sep 05 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..325
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Cf. A024429, A024430, A121867, A121868, A121869.

List([0..25], n-> (Sum([0..Int(n/2)], k-> Stirling2(n,2*k)*(-1)^(k)) )^2 + (Sum([0..Int(n/2)], k-> (-1)^k*Stirling2(n,2*k+1)))^2 ); # G. C. Greubel, Oct 08 2019

C:= ComplexField(); a:= func< n | Round(Abs( (&+[I^k*StirlingSecond(n,k): k in [0..n]])^2 )) >;
[a(n): n in [0..25]]; // G. C. Greubel, Oct 08 2019

Maple

A121870a:= proc(a) local i, t: i:=1: t:=0: for i to 100 do t:=t+evalf((i^(a-1))*(I)^i/(i)!): od: RETURN(round(abs(t^2))): end: a:= A121870a(n); # Russell Walsmith, Apr 18 2008 # Alternate: seq(abs(BellB(n,I))^2, n=0..30); # Robert Israel, Oct 15 2017

Mathematica

Table[Abs[BellB[n, I]]^2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2017 *)

PARI

a(n) = abs( (sum(k=0,n, I^k*stirling(n,k,2)))^2 ); vector(25, n, a(n-1)) \\ G. C. Greubel, Oct 08 2019

Sage

[abs( sum(I^k*stirling_number2(n,k) for k in (0..n))^2 ) for n in (0..25)] # G. C. Greubel, Oct 08 2019

Formula

a(n) = A121867(n)^2 + A121868(n)^2.

From Gary W. Adamson, Jul 22 2011: (Start)

sqrt(a(n)) = upper left term in M^n as to the modulus of a polar term; M = an infinite square production matrix in which a column of (i, i, i, ...) is appended to the right of Pascal's triangle, as follows (with i = sqrt(-1)):

1, i, 0, 0, 0, ...

1, 1, i, 0, 0, ...

1, 2, 1, i, 0, ...

1, 3, 3, 1, i, ...

... (End)

a(n) = |B_n(i)|^2, where B_n(x) is the n-th Bell polynomial, i = sqrt(-1) is the imaginary unit. - Vladimir Reshetnikov, Oct 15 2017

a(n) ~ (n*exp(-1 + Re(LambertW(i*n)) / Abs(LambertW(i*n))^2) / Abs(LambertW(i*n)))^(2*n) / Abs(1 + LambertW(i*n)), where i is the imaginary unit. - Vaclav Kotesovec, Jul 28 2021

cp's OEIS Frontend

A357728 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * (exp(x) - 1) ).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A357725 Expansion of e.g.f. cos( sqrt(2) * (exp(x) - 1) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

A121869 Monthly Problem 10791, first expression.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A121870 Monthly Problem 10791, second expression.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula