A241171 -id:A241171 - cp's OEIS frontend

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

1, 0, 1, 0, 1, 20, 0, 1, 168, 1680, 0, 1, 1364, 55440, 369600, 0, 1, 10920, 1561560, 33633600, 168168000, 0, 1, 87380, 42771456, 2385102720, 34306272000, 137225088000, 0, 1, 699048, 1160164320, 158411809920, 5105916816000, 54752810112000, 182509367040000

Offset: 0

Peter Luschny, Jan 22 2017

			Triangle begins:
[1]
[0, 1]
[0, 1,    20]
[0, 1,   168,    1680]
[0, 1,  1364,   55440,   369600]
[0, 1, 10920, 1561560, 33633600, 168168000]

Cf. A014606 (diagonal), A243664 (row sums), A002115 (alternating row sums), A281479 (central coefficients), A327023 (refinement).

Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278074 (m=4).

P := proc(m, n) option remember; if n = 0 then 1 else
add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
for n from 0 to 6 do PolynomialTools:-CoefficientList(P(3,n), x) od;
# Alternatively:
A278073_row := proc(n)
1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1));
expand(series(%,x,3*n+1)); (3*n)!*coeff(%,x,3*n);
PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 6 do A278073_row(n) od;

With[{m = 3}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 21, m}]];
Function[arg, CoefficientList[arg, t]] /@ % // Flatten

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(30)
@cached_function
def P(m, n):
    if n == 0: return R(1)
    return expand(sum(binomial(m*n, m*k)*P(m, n-k)*x for k in (1..n)))
def A278073_row(n): return list(P(3, n))
for n in (0..6): print(A278073_row(n)) # Peter Luschny, Mar 24 2020

E.g.f.: 1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1)), nonzero terms.

A210657 a(0)=1; thereafter a(n) = -2Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, -2, 22, -602, 30742, -2523002, 303692662, -50402079002, 11030684333782, -3077986048956602, 1066578948824962102, -449342758735568563802, 226182806795367665865622, -134065091768709178087428602, 92423044260377387363207812342, -73323347841467639992211297199002

Offset: 0

N. J. A. Sloane, Mar 28 2012

sign

The version without signs has an interpretation as a sum over marked Schröder paths. See the Josuat-Verges and Kim reference.
Consider the sequence defined by a(0)=1; thereafter a(n) = c*Sum_{k=1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
Apparently a(n) = 2*(-1)^n*A002114(n). - R. J. Mathar, Mar 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..200
Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO], 2011.
Zhi-Hong Sun, On the further properties of U_n, arXiv:1203.5977 [math.NT], 2012.

Cf. A002114, A028296, A094088, A210657, A210672, A210674, A210676, A249939, A255882, A241171.

A210657:=proc(n) option remember;
   if n=0 then 1
   else -2*add(binomial(2*n,2*k)*procname(n-k),k=1..floor(n)); fi;
end;
[seq(f(n),n=0..20)];
# Second program:
a := (n) -> 2*36^n*(Zeta(0,-n*2,1/6)-Zeta(0,-n*2,2/3)):
seq(a(n), n=0..15); # Peter Luschny, Mar 11 2015

nmax=20; Table[(CoefficientList[Series[1/(2*Cosh[x]-1), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)
Table[9^n EulerE[2 n, 1/3], {n, 0, 20}] (* Vladimir Reshetnikov, Jun 05 2016 *)

a(n)=polcoeff(sum(m=0, n, (2*m)!*(-x)^m/prod(k=1, m, 1-k^2*x +x*O(x^n)) ), n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Sep 17 2012

O.g.f.: Sum_{n>=0} (2*n)! * (-x)^n / Product_{k=1..n} (1 - k^2*x). - Paul D. Hanna, Sep 17 2012
E.g.f.: 1/(2*cosh(x) - 1) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!. - Paul D. Hanna, Oct 30 2014
E.g.f.: cos(z/2)/cos(3z/2) = Sum_{n>=0} abs(a(n))*x^(2*n)/(2*n)!. - Olivier Gérard, Feb 12 2014
From Peter Bala, Mar 09 2015: (Start)
a(n) = 3^(2*n)*E(2*n,1/3), where E(n,x) is the n-th Euler polynomial.
O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - x*(3*k + 1)^2).
O.g.f. as a continued fraction: 1/(1 + (3^2 - 1^2)*x/(4 + 12^2*x/(4 + (18^2 - 2^2)*x/(4 + 24^2*x/(4 + (30^2 - 2^2)*x/(4 + 36^2*x/(4 + ... ))))))) = 1 - 2*x + 22*x^2 - 602*x^3 + 30742*x^4 - .... See Josuat-Vergès and Kim, p. 23.
The expansion of exp( Sum_{n >= 1} a(n)*x^n/n ) appears to have integer coefficients. See A255882. (End)
a(n) = 2*36^n*(zeta(-n*2,1/6)-zeta(-n*2,2/3)), where zeta(a,z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
a(n) ~ 2 * (-1)^n * (2*n)! * 3^(2*n+1/2) / Pi^(2*n+1). - Vaclav Kotesovec, Mar 14 2015
a(n) = Sum_{k=0..n} A241171(n, k)*(-2)^k. - Peter Luschny, Sep 03 2022

A278074 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 4.

Original entry on oeis.org

1, 0, 1, 0, 1, 70, 0, 1, 990, 34650, 0, 1, 16510, 2702700, 63063000, 0, 1, 261630, 213519150, 17459442000, 305540235000, 0, 1, 4196350, 17651304000, 4350310965000, 231905038365000, 3246670537110000

Offset: 0

Peter Luschny, Jan 22 2017

			Triangle starts:
[1]
[0, 1]
[0, 1,     70]
[0, 1,    990,     34650]
[0, 1,  16510,   2702700,    63063000]
[0, 1, 261630, 213519150, 17459442000, 305540235000]

Robert Israel, Table of n, a(n) for n = 0..8510

Cf. A014608 (diagonal), A243665 (row sums), A211212 (alternating row sums), A281480 (central coefficients).
Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278073 (m=3).
Cf. A327024 (refinement).

P := proc(m,n) option remember; if n = 0 then 1 else
add(binomial(m*n,m*k)* P(m,n-k)*x, k=1..n) fi end:
for n from 0 to 6 do PolynomialTools:-CoefficientList(P(4,n), x) od;
# Alternatively:
A278074_row := proc(n) 1/(1-t*((cosh(x)+cos(x))/2-1)); expand(series(%,x,4*n+1));
(4*n)!*coeff(%,x,4*n); PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 5 do A278074_row(n) od;

With[{m = 4}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 24, m}]];
Function[arg, CoefficientList[arg, t]] /@ % // Flatten

# uses [P from A278073]
def A278074_row(n): return list(P(4, n))
for n in (0..6): print(A278074_row(n)) # Peter Luschny, Mar 24 2020

E.g.f.: 1/(1-t*((cosh(x)+cos(x))/2-1)), nonzero terms.

A210672 a(0)=1; thereafter a(n) = 2Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, 2, 26, 842, 50906, 4946282, 704888186, 138502957322, 35887046307866, 11855682722913962, 4863821092813045946, 2425978759725443056202, 1445750991051368583278426, 1014551931766896667943384042, 828063237870027116855857421306, 777768202388460616924079724057482

Offset: 0

N. J. A. Sloane, Mar 28 2012

Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k = 1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255929. - Peter Bala, Mar 13 2015
The Stirling-Bernoulli transform of Fibonacci(n+1) = 1, 1, 2, 3, 5, 8, 13, ... is 1, 0, 2, 0, 26, 0, 842, 0, 50906, 0, ... - Philippe Deléham, May 25 2015

G. C. Greubel, Table of n, a(n) for n = 0..220
Hacène Belbachir, Yahia Djemmada, On central Fubini-like numbers and polynomials, arXiv:1811.06734 [math.CO], 2018. See p. 4.

Cf. A028296, A094088, A210657, A210674, A210676, A255929, A241171.

f:=proc(n,k) option remember;  local i;
if n=0 then 1
else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
g:=k->[seq(f(n,k),n=0..40)];
g(2);

nmax=20; Table[(CoefficientList[Series[1/(3-2*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)

a(n) ~ 2*sqrt(Pi/5) * n^(2*n+1/2) / (exp(2*n) * (log((1+sqrt(5))/2))^(2*n+1)). - Vaclav Kotesovec, Mar 13 2015
E.g.f.: 1/(3-2*cosh(x)) (even coefficients). - Vaclav Kotesovec, Mar 14 2015
a(n) = Sum_{k = 0..2*n} A163626(2*n,k)*A000045(n+1). - Philippe Deléham, May 25 2015
a(n) = Sum_{k=0..n} A241171(n, k)*2^k. - Peter Luschny, Sep 03 2022

A002446 a(n) = 2^(2*n+1) - 2.

Original entry on oeis.org

0, 6, 30, 126, 510, 2046, 8190, 32766, 131070, 524286, 2097150, 8388606, 33554430, 134217726, 536870910, 2147483646, 8589934590, 34359738366, 137438953470, 549755813886, 2199023255550, 8796093022206, 35184372088830

Offset: 0

N. J. A. Sloane

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Equals 6 * A002450(n).

A diagonal of the triangle in A241171.

List([0..30], n-> 2*(4^n - 1)) # G. C. Greubel, Jul 04 2019

[2^(2*n+1) - 2: n in [0..30]]; // Vincenzo Librandi, Jun 01 2011

A002446:=6*z/((4*z-1)*(z-1)); # [Generating function. Simon Plouffe in his 1992 dissertation.]

f[n_] := Det[{{1, 1}, {1, 4}}^(n - 1) {{1, 2}, {1, 2}}]; Array[f, 30] (* Robert G. Wilson v, Jul 13 2011 *)
2^(2*Range[0,30]+1)-2 (* or *) LinearRecurrence[{5,-4},{0,6},30] (* Harvey P. Dale, Sep 01 2016 *)

a(n) = 2*(4^n - 1); \\ G. C. Greubel, Jul 04 2019

[2*(4^n -1) for n in (0..30)] # G. C. Greubel, Jul 04 2019

G.f.: 6*x/((1-x)*(1-4*x)). - Simon Plouffe, see MAPLE line

E.g.f.: (cos(i*x)-1)^2. - Vladimir Kruchinin, Oct 28 2012

More terms from Vincenzo Librandi, Jun 01 2011

A100872 a(n) = (1/sqrt(5)) * Sum_{k>0} k^(2n)/phi^(2k) where phi = (1+sqrt(5))/2 = A001622.

Original entry on oeis.org

1, 13, 421, 25453, 2473141, 352444093, 69251478661, 17943523153933, 5927841361456981, 2431910546406522973, 1212989379862721528101, 722875495525684291639213, 507275965883448333971692021, 414031618935013558427928710653, 388884101194230308462039862028741

Offset: 1

Benoit Cloitre, Jan 08 2005

A bisection of "Stirling-Bernoulli transform" of Fibonacci numbers.

Alois P. Heinz, Table of n, a(n) for n = 1..223

Cf. A001622, A050946, A100868, A241171.

FullSimplify[Table[PolyLog[-2k, GoldenRatio^(-2)]/Sqrt[5], {k, 1, 10}]] (* Vladimir Reshetnikov, Feb 16 2011 *)
T[n_, k_] /; 1 <= k <= n := T[n, k] = k(2k-1) T[n-1, k-1] + k^2 T[n-1, k]; T[, 1] = 1; T[, ] = 0; a[n] := Sum[2^(k-1) T[n, k], {k, 1, n}]; Array[a, 15] (* Jean-François Alcover, Jul 03 2019 *)

a(n)=round(1/sqrt(5)*sum(k=1,500,k^(2*n)/((1+sqrt(5))/2)^(2*k)))

a(n) = A050946(2*n).
From Peter Bala, Aug 20 2014: (Start)
E.g.f.: -1/2 + (1/2)*exp(z)/(3*exp(z) - exp(2*z) - 1) = z^2/2! + 13*z^4/4! + 421*z^6/6! + ....
a(n) = Sum_{k = 1..n} 2^(k-1)*A241171(n,k).
a(n) = Sum_{1 <= j <= k <= n} (-1)^(k-j)*binomial(2*k,k+j)*j^(2*n).
(End)

A255926 Expansion of exp( Sum_{n >= 1} A210676(n)*x^n/n ).

Original entry on oeis.org

1, -3, 30, -802, 45414, -4508190, 692197470, -151610017950, 44827810930305, -17193060505570335, 8298004578522898140, -4920774627129981351120, 3516683319021255757053900, -2980761698101283167670391780, 2956463734237276273792194346560, -3392220222832838757465019626175680

Offset: 0

Peter Bala, Mar 11 2015

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 - 3*x + 30*x^2 - 802*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A210676.
This sequence is the particular case m = -3 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
For cases see A255882(m = -2), A255881(m = -1), A255928(m = 1), A255929(m = 2) and A255930(m = 3).
Note that u(n), as a polynomial in the variable m, is the n-th row polynomial of A241171.

Cf. A210676, A241171, A255882(m = -2), A255881(m = -1), A255928(m = 1), A255929(m = 2), A255930(m = 3).

A210676 := proc (n) option remember; if n = 0 then 1 else -3*add(binomial(2*n, 2*k)*A210676(k), k = 0 .. n-1) end if; end proc:
A255926 := proc (n) option remember; if n = 0 then 1 else add(A210676(n-k)*A255926(k), k = 0 .. n-1)/n end if; end proc:
seq(A255926(n), n = 0 .. 16);

O.g.f.: exp(-3*x + 51*x^2/2 - 2163*x^3/3 + 171231*x^4/4 + ...) = 1 - 3*x + 30*x^2 - 802*x^3 + 45414*x^4 - ....

a(0) = 1 and a(n) = (1/n)*Sum_{k = 0..n-1} A210676(n-k)*a(k) for n >= 1.

A255928 Expansion of exp( Sum_{n >= 1} A094088(n)*x^n/n ).

Original entry on oeis.org

1, 1, 4, 44, 1025, 41693, 2617128, 234091692, 28251572652, 4421489003700, 870650503128708, 210629395976568828, 61405707768736724472, 21231253444779700476672, 8589776776743377081599500, 4020181599664131540547091076, 2155088041310451318611119556661

Offset: 0

Peter Bala, Mar 11 2015

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 + x + 4*x^2 + 44*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A094088.
This sequence is the particular case m = 1 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
For cases see A255926(m = -3), A255882(m = -2), A255881(m = -1), A255929(m = 2) and A255930(m = 3).
Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.

Cf. A094088, A241171, A255926(m = -3), A255882(m = -2), A255881(m = -1), A255929(m = 2), A255930(m = 3).

A094088 := proc (n) option remember; if n = 0 then 1 else add(binomial(2*n, 2*k)*A094088(k), k = 0 .. n-1) end if; end proc:
A255928 := proc (n) option remember; if n = 0 then 1 else add(A094088(n-k)*A255928(k), k = 0 .. n-1)/n end if; end proc:
seq(A255928(n), n = 0 .. 16);

O.g.f.: exp(x + 7*x^2/2 + 121*x^3/3 + 3907*x^4/4 + ...) = 1 + x + 4*x^2 + 44*x^3 + 1025*x^4 + ....

a(0) = 1 and a(n) = (1/n)*Sum_{k = 0..n-1} A094088(n-k)*a(k) for n >= 1.

A255929 Expansion of exp( Sum_{n >= 1} A210672(n)*x^n/n ).

Original entry on oeis.org

1, 2, 15, 308, 13399, 1019106, 119698377, 20039968920, 4527610159068, 1326616296092984, 489092182592254708, 221537815033845709776, 120928125204565597029220, 78286897353506845258973144, 59305342759674536454338570652, 51970719684035315747385128783808

Offset: 0

Peter Bala, Mar 11 2015

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 + 2*x + 15*x^2 + 308*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A210672.
This sequence is the particular case m = 2 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
For cases see A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928(m = 1) and A255930(m = 3).
Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.

A210672, A241171, A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928(m = 1), A255930(m = 3).

#A255929
A210672 := proc (n) option remember; if n = 0 then 1 else 2*add(binomial(2*n, 2*k)*A210672(k), k = 0 .. n-1) end if; end proc:
A255929 := proc (n) option remember; if n = 0 then 1 else add(A210672(n-k)*A255929(k), k = 0 .. n-1)/n end if; end proc:
seq(A255929(n), n = 0 .. 15);

O.g.f.: exp(2*x + 26*x^2/2 + 842*x^3/3 + 50906*x^4/4 + ...) = 1 + 2*x + 15*x^2 + 308*x^3 + 13399*x^4 + ....

a(0) = 1 and a(n) = 1/n*Sum_{k = 0..n-1} A210672(n-k)*a(k) for n >= 1.

A255930 Expansion of exp( Sum_{n >= 1} A210674(n)*x^n/n ).

Original entry on oeis.org

1, 3, 33, 991, 63060, 7018860, 1206748720, 295775068680, 97835325011235, 41970842737399345, 22655642596496388759, 15025240474194493147857, 12008582230377080862401692, 11382727559611560650861409564, 12625404970864692720119281536900, 16199644066580777034289339157904220

Offset: 0

Peter Bala, Mar 11 2015

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 + 3*x + 33*x^2 + 991*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A210674.
This sequence is the particular case m = 3 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
For cases see A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928 (m = 1) and A255929(m = 2).
Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.

Cf. A210674, A241171, A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928(m = 1), A255929(m = 2).

#A255930
A210674 := proc (n) option remember; if n = 0 then 1 else 3*add(binomial(2*n, 2*k)*A210674(k), k = 0 .. n-1) end if; end proc:
A255930 := proc (n) option remember; if n = 0 then 1 else add(A210674(n-k)*A255930(k), k = 0 .. n-1)/n end if; end proc:
seq(A255930(n), n = 0 .. 15);

O.g.f.: exp(3*x + 57*x^2/2 + 2703*x^3/3 + 239277*x^4/4 + ...) = 1 + 3*x + 33*x^2 + 991*x^3 + 63060*x^4 + ....

a(0) = 1 and a(n) = 1/n*Sum_{k = 0..n-1} A210674(n-k)*a(k) for n >= 1.

cp's OEIS Frontend

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A210657 a(0)=1; thereafter a(n) = -2*Sum_{k=1..n} binomial(2n,2k)*a(n-k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A278074 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)* P(m, n-k)*z with P(m, 0) = 1 and m = 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A210672 a(0)=1; thereafter a(n) = 2*Sum_{k=1..n} binomial(2n,2k)*a(n-k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A002446 a(n) = 2^(2*n+1) - 2.

Views

Author

Keywords

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A100872 a(n) = (1/sqrt(5)) * Sum_{k>0} k^(2n)/phi^(2k) where phi = (1+sqrt(5))/2 = A001622.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A255926 Expansion of exp( Sum_{n >= 1} A210676(n)*x^n/n ).

Views

Author

Keywords

Comments

Crossrefs

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

A210657 a(0)=1; thereafter a(n) = -2Sum_{k=1..n} binomial(2n,2k)a(n-k).

A278074 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 4.

A210672 a(0)=1; thereafter a(n) = 2Sum_{k=1..n} binomial(2n,2k)a(n-k).