cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A116383 Row sums of number triangle A116382.

Original entry on oeis.org

1, 1, 4, 6, 19, 33, 93, 175, 460, 910, 2286, 4676, 11388, 23842, 56808, 120926, 283611, 611065, 1416625, 3079635, 7078263, 15490553, 35374519, 77805481, 176813809, 390379483, 883861033, 1957097715, 4418562265, 9805546875, 22090136885
Offset: 0

Views

Author

Paul Barry, Feb 12 2006

Keywords

Comments

Binomial transform is A116387.

Links

Programs

  • GAP
    List([0..40], n-> Sum([0..n], k-> Sum([0..n], j-> (-1)^(n-j)* Binomial(n,j)*Sum([0..j], m-> Binomial(j,m-k)*Binomial(m,j-m) )))); # G. C. Greubel, May 22 2019
  • Magma
    T:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*(&+[Binomial(j,m-k)* Binomial(m,j-m): m in [0..j]]): j in [0..n]]) >;
[(&+[T(n,k): k in [0..n]]): n in [0..40]]; // G. C. Greubel, May 22 2019
  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1+x-4*x^2)/(2*(1-5*x^2)*Sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2)) )); // G. C. Greubel, May 22 2019
  • Mathematica
    CoefficientList[Series[(1+x-4*x^2)/(2*(1-5*x^2)*Sqrt[1-4*x^2])+(1+x)/(2*(1-5*x^2)), {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
  • PARI
    my(x='x+O('x^40)); Vec((1+x-4*x^2)/(2*(1-5*x^2)*sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2))) \\ G. C. Greubel, May 22 2019
  • Sage
    ((1+x-4*x^2)/(2*(1-5*x^2)*sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 22 2019

Formula

G.f.: (1+x-4*x^2)/(2*(1-5*x^2)*sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2)).
a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n,j)*Sum_{i=0..j} C(j,i-k) * C(i,j-i).
Conjecture: n*a(n) + (n-2)*a(n-1) + (-13*n+20)*a(n-2) + (-9*n+22)*a(n-3) + 4*(14*n-45)*a(n-4) + 20*(n-3)*a(n-5) + 80*(-n+5)*a(n-6) = 0. - R. J. Mathar, Nov 24 2012
Recurrence: n*(n^2 - 4*n - 1)*a(n) = 4*(2*n-3)*a(n-1) + (9*n^3 - 40*n^2 + 3*n + 48)*a(n-2) - 20*(2*n-3)*a(n-3) - 20*(n-3)*(n^2 - 2*n - 4)*a(n-4). - Vaclav Kotesovec, Feb 12 2014
a(n) ~ (5+sqrt(5))/10 * 5^(n/2). - Vaclav Kotesovec, Feb 12 2014

A116384 Diagonal sums of the Riordan array A116382.

Original entry on oeis.org

1, 0, 3, 1, 10, 6, 36, 28, 135, 121, 517, 507, 2003, 2093, 7815, 8569, 30634, 34902, 120480, 141664, 475002, 573574, 1876294, 2318010, 7422676, 9354540, 29400192, 37708672, 116567356, 151868100, 462561572, 611180252, 1836843591, 2458123705
Offset: 0

Views

Author

Paul Barry, Feb 12 2006

Keywords

Links

Programs

  • GAP
    List([0..40], n-> Sum([0..n], k-> Sum([0..n-k], j-> (-1)^(n-k-j)*Binomial(n-k,j)*Sum([0..j], m-> Binomial(j,m-k)*Binomial(m,j-m) )))); # G. C. Greubel, May 22 2019
  • Magma
    T:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*(&+[Binomial(j,m-k)* Binomial(m,j-m): m in [0..j]]): j in [0..n]]) >;
[(&+[T(n-k,k): k in [0..Floor(n/2)]]): n in [0..40]];
  • Mathematica
    T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[n, j]*Sum[Binomial[j, i-k]* Binomial[i, j-i], {i, 0, j}], {j, 0, n}]; Table[Sum[T[n-k, k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* G. C. Greubel, May 22 2019 *)
  • PARI
    {T(n,k) = sum(j=0,n, (-1)^(n-j)*binomial(n,j)*sum(m=0,j, binomial(j,m-k)*binomial(m,j-m) ))};vector(40, n, n--; sum(k=0, floor(n/2), T(n-k,k)) ) \\ G. C. Greubel, May 22 2019
  • Sage
    def T(n, k): return sum((-1)^(n-j)*binomial(n,j)*sum(binomial(j,m-k)*binomial(m,j-m) for m in (0..j)) for j in (0..n))
[ sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..40)] # G. C. Greubel, May 22 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} (-1)^(n-k-j)*C(n-k,j) * Sum_{i=0..j} C(j,i-k)C(i,j-i).

A114422 Riordan array (1/sqrt(1-2*x-3*x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 7, 9, 5, 1, 19, 26, 19, 7, 1, 51, 75, 65, 33, 9, 1, 141, 216, 211, 132, 51, 11, 1, 393, 623, 665, 483, 235, 73, 13, 1, 1107, 1800, 2058, 1674, 963, 382, 99, 15, 1, 3139, 5211, 6294, 5598, 3663, 1739, 581, 129, 17, 1, 8953, 15115, 19095, 18261, 13243
Offset: 0

Views

Author

Paul Barry, Feb 12 2006

Keywords

Comments

First column is central trinomial numbers A002426.
Second column is A005774.
Third column is A025568.
Row sums are A116387.
Diagonal sums are A116388.
Product of A007318 and A116382.
Column k has e.g.f. exp(x)*Sum_{j=0..k} C(k,j)*Bessel_I(k+j,2*x).

Examples

			Triangle begins
1,
1, 1,
3, 3, 1,
7, 9, 5, 1,
19, 26, 19, 7, 1,
51, 75, 65, 33, 9, 1,
141, 216, 211, 132, 51, 11, 1

Links

Programs

  • GAP
    T:=Flat(List([0..10], n->List([0..n], k->Sum([0..n], j-> Binomial(n, j-k)*Binomial(j, n-j))))); # G. C. Greubel, Dec 15 2018
  • Magma
    [[(&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Dec 15 2018
  • Mathematica
    T[n_, k_] := Sum[Binomial[n, j - k]*Binomial[j, n - j], {j,0,n}]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)
  • PARI
    {T(n,k) = sum(j=0,n, binomial(n, j-k)*binomial(j, n-j))};
for(n=0, 10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 15 2018
  • Sage
    [[sum(binomial(n, j-k)*binomial(j, n-j) for j in range(n+1)) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Dec 15 2018

Formula

Riordan array (1/sqrt(1-2*x-3*x^2), (1-x-2*x^2-sqrt(1-2*x-3*x^2) ) / (2*x^2)).
Number triangle T(n,k) = Sum_{j=0..n} C(n,j-k)*C(j,n-j).

A114453 Number of 5-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 4, 76, 963, 11185, 124465, 1349779, 14371023, 150982388, 1570678136, 16218372618, 166497674684, 1701439985694, 17323079621014, 175846040834673, 1780617141307093, 17993699600756449, 181520864946969233
Offset: 0

Views

Author

Robert G. Wilson v, Feb 07 2006

Keywords

Examples

			There are 4 five-almost primes up to 100: 32,48,72 and 80, so a(2) = 4.

Crossrefs

Cf. A006880, A066265, A109251, A014614, A116382.

Programs

  • Mathematica
    FiveAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k*Prime@l)] - l + 1, {i, PrimePi[n^(1/5)]}, {j, i, PrimePi[(n/Prime@i)^(1/4)]}, {k, j, PrimePi[(n/(Prime@i*Prime@j))^(1/3)]}, {l, k, PrimePi[(n/(Prime@i*Prime@j*Prime@k))^(1/2)]}]; Table[ FiveAlmostPrimePi[10^n], {n, 0, 12}]
  • Python
    from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A114453(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,5))) # Chai Wah Wu, Sep 18 2024

Extensions

a(13)-a(14) from Robert G. Wilson v, Jan 07 2007
a(15)-a(18) from Henri Lifchitz, Feb 03 2025

A120049 Number of 8-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 7, 105, 1418, 17572, 207207, 2367507, 26483012, 291646797, 3173159326, 34192782745, 365561221293, 3882841742380, 41015564702074, 431227959019552, 4515480975731045, 47115876816676830
Offset: 0

Views

Author

Robert G. Wilson v, Feb 07 2006

Keywords

Examples

			There are 7 eight-almost primes up to 1000: 256, 384, 576, 640, 864, 896 & 960.

Crossrefs

Cf. A066265, A014613, A116382.
Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[8, 10^n], {n, 12}]
  • Python
    from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A120049(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,8))) # Chai Wah Wu, Aug 23 2024

Extensions

a(13)-a(14) from Robert G. Wilson v, Jan 07 2007
Example corrected by Harvey P. Dale, Aug 13 2018
a(15)-a(18) from Henri Lifchitz, Mar 18 2025

A120047 Number of 6-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 2, 37, 485, 5933, 68963, 774078, 8493366, 91683887, 977694273, 10327249593, 108264085934, 1128049914377, 11694704489580, 120734708167792, 1242063105505230, 12739510126065301, 130330025583399801
Offset: 0

Views

Author

Robert G. Wilson v, Feb 07 2006

Keywords

Examples

			There are 2 six-almost primes up to 100: 64 and 96, so a(2) = 2.

Crossrefs

Cf. A066265, A014613, A116382.
Cf. A006880, A036352, A109251, A114106, A114453.
Cf. A120048, A120049, A120050, A120051, A120052, A120053, A116430.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[6, 10^n], {n, 0, 13}]
  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def almostprimepi(n,k):
    if k==0: return int(n>=1)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,k)) if k>1 else primepi(n))
def A120047(n): return almostprimepi(10**n,6) # Chai Wah Wu, Dec 09 2024

Extensions

a(14) from Robert G. Wilson v, Jan 07 2007
a(15)-a(18) from Henri Lifchitz, Feb 03 2025

A120048 Number of 7-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 14, 231, 2973, 35585, 409849, 4600247, 50678212, 550454756, 5913771637, 62981797962, 665997804082, 7001087934965, 73232029374751, 762783057783010, 7916319351632036, 81898808371556517
Offset: 0

Views

Author

Robert G. Wilson v, Feb 07 2006

Keywords

Examples

			There are 14 seven-almost primes up to 1000: 128, 192, 288, 320, 432, 448, 480, 648, 672, 704, 720, 800, 832 & 972.

Crossrefs

Cf. A066265, A014613, A116382.
Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[7, 10^n], {n, 11}]

Extensions

More terms from Robert G. Wilson v, Jan 07 2007
Example corrected by Harvey P. Dale, Jan 25 2013
a(15)-a(18) from Henri Lifchitz, Mar 18 2025

A120050 Number of 9-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 2, 47, 671, 8491, 101787, 1180751, 13377156, 148930536, 1636170477, 17787688377, 191742524399, 2052389350029, 21838745177567, 231206458686127, 2437121982958248, 25591920108631224, 267840642082525459
Offset: 0

Views

Author

Robert G. Wilson v, Feb 07 2006

Keywords

Examples

			There are 2 nine-almost primes up to 1000: 512 & 768.

Crossrefs

Cf. A066265, A014613, A116382.
Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[9, 10^n], {n, 12}]

Extensions

a(13) and a(14) from Robert G. Wilson v, Jan 07 2007
a(15)-a(19) from Henri Lifchitz, Mar 18 2025

A120051 Number of 10-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 0, 22, 306, 4016, 49163, 578154, 6618221, 74342563, 823164388, 9011965866, 97765974368, 1052666075366, 11263041623194, 119864659464824, 1269754732725522, 13396817167474205, 140847445420555406
Offset: 0

Views

Author

Robert G. Wilson v, Feb 07 2006

Keywords

Examples

			There are 22 ten-almost primes up to 10000: 1024, 1536, 2304, 2560, 3456, 3584, 3840, 5184, 5376, 5632, 5760, 6400, 6656, 7776, 8064, 8448, 8640, 8704, 8960, 9600, 9728, and 9984.

Crossrefs

Cf. A066265, A014613, A116382.
Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[10, 10^n], {n, 12}]
  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A120051(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,10))) # Chai Wah Wu, Nov 03 2024

Extensions

More terms from Robert G. Wilson v, Jan 07 2007
a(15)-a(19) from Henri Lifchitz, Mar 20 2025

A120053 Number of 12-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 0, 3, 63, 865, 11068, 133862, 1563465, 17836903, 200051717, 2214357712, 24255601105, 263439785143, 2841076717752, 30457549169277, 324855769153426, 3449587218984911, 36489283363168885
Offset: 0

Views

Author

Robert G. Wilson v, Feb 07 2006

Keywords

Examples

			There are 3 twelve-almost primes up to 10000: 4096, 6144, and 9216.

Crossrefs

Cf. A066265, A014613, A116382.
Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[12, 10^n], {n, 11}]
  • Python
    from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A120053(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,12))) # Chai Wah Wu, Aug 23 2024

Extensions

a(13) and a(14) from Robert G. Wilson v, Jan 07 2007
a(15) from Chai Wah Wu, Aug 24 2024
a(16)-a(19) from Henri Lifchitz, Mar 18 2025
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