Prabha Sivaramannair - cp's OEIS frontend

A377728 Convolution of Leonardo numbers with Jacobsthal numbers.

0, 1, 2, 7, 16, 39, 86, 189, 402, 847, 1760, 3631, 7438, 15165, 30794, 62343, 125904, 253783, 510758, 1026685, 2061730, 4136991, 8295872, 16627167, 33311646, 66716029, 133582106, 267406999, 535206832, 1071049287, 2143127030, 4287918141, 8578528818, 17161414255

Offset: 0

Prabha Sivaramannair, Nov 05 2024

Prabha Sivaraman Nair, Convolution Identities of p-numbers, Math. Montis. (2024), pp. 26-43.
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,1,2).

Cf. A001045, A001595.

LinearRecurrence[{3, 0, -5, 1, 2}, {0, 1, 2, 7, 16}, 34] (* Amiram Eldar, Nov 07 2024 *)

from sympy import fibonacci
def A377728(n): return 1-(fibonacci(n+2)<<2)+(m:=(4<>1 # Chai Wah Wu, Nov 09 2024

a(n) = Sum_{i=0..n} L(i)*J(n-i) where L = A001595 and J = A001045.
a(n) = (3*J(n+2) - 2*L(n+1) - 1)/2 where L = A001595 and J = A001045.
G.f.: -x*(x^2-x+1)/((x-1)*(2*x-1)*(x+1)*(x^2+x-1)). - Alois P. Heinz, Nov 05 2024
E.g.f.: 2*cosh(2*x) + sinh(x) + 2*sinh(2*x) - 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Nov 06 2024

A372738 Binomial transform of A369795.

Original entry on oeis.org

1, 4, 28, 298, 4240, 75394, 1608688, 40045618, 1139279680, 36463487554, 1296712045648, 50724943433938, 2164652356532320, 100072984472662114, 4982304066392196208, 265770533884409878258, 15122101633293034668160, 914210942121577873619074, 58519992421072004957876368, 3954059527570115477197922578

Offset: 0

Prabha Sivaramannair, May 11 2024

nonn

Cf. A369795, A355408.

nmax = 20; CoefficientList[Series[E^(2*x)/(1 + E^x - E^(3*x)), {x, 0, nmax}], x]*Range[0, nmax]! (* Vaclav Kotesovec, Jun 01 2024 *)

def a(n):
    if n==0:
        return 1
    else:
        return sum([(1-(-1)^j-(-2)^j)*binomial(n,j)*a(n-j) for j in [1,..,n]])
list(a(n) for n in [0,..,20])

a(n) = Sum_{j=1..n} (1-(-1)^j-(-2)^j)*binomial(n,j)*a(n-j) for n > 0.
a(n) = 2^n + Sum_{j=1..n} (3^j-1)*binomial(n,j)*a(n-j).
a(n) = 1 + Sum_{j=1..n} (2^j-(-1)^j)*binomial(n,j)*a(n-j).
E.g.f.: exp(2*x)/(1 + exp(x) - exp(3*x)). - Vaclav Kotesovec, Jun 01 2024

A371984 Binomial transform of A371460.

Original entry on oeis.org

1, 3, 15, 117, 1227, 16053, 251955, 4613997, 96566667, 2273672133, 59482039395, 1711735382877, 53737315411707, 1827584253650613, 66936582030410835, 2626714554845111757, 109948916113808074347, 4889877314768678051493

Offset: 0

Prabha Sivaramannair, Apr 15 2024

nonn

Cf. A371460.

nn = 17; a[0] = 1; Do[Set[a[n], 2^n + Sum[(3^j - 2^j)*Binomial[n, j]*a[n - j], {j, n}]], {n, nn}]; Array[a, nn + 1, 0] (* Michael De Vlieger, Apr 19 2024 *)

def a(n):
    if n==0:
        return 1
    else:
        return sum([(1-(-2)^j)*binomial(n,j)*a(n-j) for j in [1,..,n]])
list(a(n) for n in [0,..,20])

f= e^(2*x)/(1 + e^(2*x) - e^(3*x))
print([(diff(f,x,i)).subs(x=0) for i in [0,..,20]])

a(0) = 1, a(n) = Sum_{j=1..n} (1-(-2)^j)*binomial(n,j)*a(n-j) for n > 0.
a(0) = 1, a(n) = 2^n + Sum_{j=1..n} (3^j-2^j)*binomial(n,j)*a(n-j) for n > 0.
E.g.f.: exp(2*x)/(1 + exp(2*x) - exp(3*x)).

A371460 Binomial transform of A355409.

Original entry on oeis.org

1, 2, 10, 80, 838, 10952, 171910, 3148280, 65890198, 1551389192, 40586247910, 1167964662680, 36666464437558, 1247011549249832, 45672691012357510, 1792280373542404280, 75021202465129000918, 3336499249170658956872, 157116438405334017308710, 7809681380575733223237080, 408621675981135189773468278

Offset: 0

Prabha Sivaramannair, Mar 24 2024

nonn

Cf. A355409.

def a(n):
    if n==0:
        return 1
    else:
        return (-1)^n + sum([(1-(-2)^j)*binomial(n,j)*a(n-j) for j in [1,..,n]])
list(a(n) for n in [0,..,20])

f= e^(x)/(1 + e^(2*x) - e^(3*x))
print([(diff(f,x,i)).subs(x=0) for i in [0,..,20]])

a(0) = 1, a(n) = (-1)^n + Sum_{j=1..n} (1-(-2)^j)*binomial(n,j)*a(n-j) for n > 0.
a(0) = 1, a(n) = 1 + Sum_{j=1..n} (3^j-2^j)*binomial(n,j)*a(n-j) for n > 0.
E.g.f.: exp(x)/(1 + exp(2*x) - exp(3*x)).

A370163 a(0) = 2, a(n) = (-1)^n + (-2)^n + (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.

Original entry on oeis.org

2, 1, 5, 25, 161, 1321, 13025, 149605, 1963841, 29004721, 475975745, 8591917885, 169193833121, 3609452038921, 82924458549665, 2041207822721365, 53594538159184001, 1495143168658285921, 44164021453758342785, 1377005070100813288045, 45193800193226286112481

Offset: 0

Prabha Sivaramannair, Feb 26 2024

nonn

Inverse binomial transform of A370092 + A370456.

Cf. A370092, A370456.

seq(n)={my(p=exp(x + O(x*x^n))); Vec(serlaplace(2*(1 + p)/(1 + p + p^2 - p^3)))} \\ Andrew Howroyd, Feb 26 2024

def a(m):
    if m==0:
        return 2
    else:
        return (-1)^m+(-2)^m+1/2*sum([(1-(-2)^j-(-1)^j)*binomial(m,j)*a(m-j) for j in [1,..,m]])
list(a(m) for m in [0,..,20])

f=2*(1+e^x)/(1+e^x+e^(2*x)-e^(3*x))
print([(diff(f,x,i)).subs(x=0) for i in [0,..,20]])

E.g.f.: 2*(1 + exp(x))/(1 + exp(x) + exp(2*x) - exp(3*x)).

A370456 a(0) = 1, a(n) = (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.

Original entry on oeis.org

1, 2, 6, 29, 192, 1577, 15516, 178229, 2339952, 34559057, 567117876, 10237161629, 201592448712, 4300618438937, 98803485774636, 2432074390036229, 63857242954421472, 1781444969999245217, 52620896463516221796, 1640684857196257578029, 53847865360369426418232

Offset: 0

Prabha Sivaramannair, Feb 23 2024

nonn

Binomial transform of A370092.

Cf. A370092, A370163.

seq(n)={my(p=exp(x + O(x*x^n))); Vec(serlaplace(2*p^2/(1 + p + p^2 - p^3)))} \\ Andrew Howroyd, Feb 23 2024

def a(m):
    if m==0:
        return 1
    else:
        return 1/2*sum([(1-(-2)^j-(-1)^j)*binomial(m,j)*a(m-j) for j in [1,..,m]])
list(a(m) for m in [0,..,20])

E.g.f.: 2*exp(2*x)/(1 + exp(x) + exp(2*x) - exp(3*x)).

A369795 Binomial transform of A355408.

Original entry on oeis.org

1, 3, 21, 225, 3201, 56913, 1214361, 30229545, 860016801, 27525472353, 978858962601, 38291126920665, 1634047719138801, 75542860973042193, 3761030066169432441, 200624240375801784585, 11415336789685550907201, 690117422445926970890433, 44175435307592982599575881

Offset: 0

Prabha Sivaramannair, Feb 01 2024

nonn

Cf. A355408.

nmax = 20; CoefficientList[Series[E^x/(1 + E^x - E^(3*x)), {x, 0, nmax}], x]*
Range[0, nmax]! (* Vaclav Kotesovec, Feb 01 2024*)

def a(m):
    if m==0:
        return 1
    else:
        return 1+sum([(3^j-1)*binomial(m,j)*a(m-j) for j in [1,..,m]])
list(a(m) for m in [1,..,50])

a(n) = 1 + Sum_{k=1..n} (3^k - 1) * binomial(n,k) * a(n-k) for n > 0.

E.g.f.: exp(x)/(1 + exp(x) - exp(3*x)). - Vaclav Kotesovec, Feb 01 2024

A370092 a(0) = 1, a(n) = (-1)^n + (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.

Original entry on oeis.org

1, 1, 3, 16, 105, 856, 8433, 96916, 1272225, 18789136, 308335713, 5565837916, 109603592145, 2338198823416, 53718370204593, 1322292130204516, 34718481333932865, 968552056638097696, 28609403248435931073, 892022330159009036716, 29276492753074019702385

Offset: 0

Prabha Sivaramannair, Feb 09 2024

nonn

Inverse binomial transform of A370456.

Conjecture: Let k > 2 be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 10 we obtain the sequence [1, 1, 3, 6, 5, 6, 3, 6, 5, 6, 3, 6, 5, 6, 3, 6, 5, 6, ...] with an apparent period of 4 beginning at a(2). See A000670 for a more general conjecture. - Peter Bala, Feb 16 2024

Cf. A370163, A370456.

a[0]=1;Table[(-1)^n+Sum[ (1-(-1)^j-  (-2) ^j) *Binomial[n,j]*a[n-j]/2,{j,1,n} ],{n,0,20}] (* James C. McMahon, Feb 10 2024 *)

seq(n)={my(p=exp(x + O(x*x^n))); Vec(serlaplace(2*p/(1 + p + p^2 - p^3)))} \\ Andrew Howroyd, Feb 10 2024

def a(m):
    if m==0:
        return 1
    else:
        return (-1)^m+1/2*sum([(1-(-2)^j-(-1)^j)*binomial(m,j)*a(m-j) for j in [1,..,m]])
list(a(m) for m in [0,..,20])

E.g.f.: 2*exp(x)/(1 + exp(x) + exp(2*x) - exp(3*x)).

cp's OEIS Frontend

User: Prabha Sivaramannair

A377728 Convolution of Leonardo numbers with Jacobsthal numbers.

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A372738 Binomial transform of A369795.

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A371984 Binomial transform of A371460.

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A371460 Binomial transform of A355409.

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A370163 a(0) = 2, a(n) = (-1)^n + (-2)^n + (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.

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A370456 a(0) = 1, a(n) = (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.

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A369795 Binomial transform of A355408.

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A370092 a(0) = 1, a(n) = (-1)^n + (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.

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Mathematica

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SageMath