A051164 -id:A051164 - cp's OEIS frontend

A051163 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a1,a1,a2,a2,a3,a3,...).

1, 2, 5, 12, 30, 76, 194, 496, 1269, 3250, 8337, 21428, 55184, 142376, 367916, 952000, 2466014, 6393372, 16586678, 43054344, 111801908, 290412296, 754543052, 1960808160, 5096293794, 13247503540, 34440553562, 89549255592, 232868582328, 605646682144

Offset: 0

Jonas Wallgren

Equals the self-convolution of A027826. Also equals antidiagonal sums of symmetric square array A100936. - Paul D. Hanna, Nov 22 2004

Equals eigensequence of triangle A152198. - Gary W. Adamson, Nov 28 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A051164, A051165, A051166, A027826, A100936, A100937, A152198.

a:= proc(n) option remember; add(`if`(k<2, 1,
      a(iquo(k, 2)))*binomial(n, k), k=0..n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 08 2015

a[n_] := a[n] = 1 + Sum[Binomial[k, j]*Binomial[n-k, j]*a[j], {k, 1, n}, {j, 0, n-k}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 11 2015 *)

a(n)=1+sum(k=1,n,sum(j=0,n-k,binomial(k,j)*binomial(n-k,j)*a(j)))

a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,(x/(1-x))^2)/(1-x)); polcoeff(A^2,n))
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Nov 22 2004

a(n) = 1 + Sum_{k=1..n} Sum_{j=0..n-k} C(k, j)*C(n-k, j)*a(j). - Paul D. Hanna, Nov 22 2004
G.f. A(x) satisfies: A(x) = A(x^2/(1-x)^2)/(1-x)^2 and A(x^2) = A(x/(1+x))/(1+x)^2. - Paul D. Hanna, Nov 22 2004
a(0) = 1; a(n) = Sum_{k=0..floor(n/2)} binomial(n+1,2*k+1) * a(k). - Ilya Gutkovskiy, Apr 07 2022

More terms from Vladeta Jovovic, Jul 26 2002

A051165 Sequence is defined by property that binomial transform of (a0,a1,a2,a3,...) = (a0,a0,a1,a1,a2,a2,a3,a3,...).

Original entry on oeis.org

1, 0, -1, 2, -4, 8, -12, 8, 15, -56, 81, -26, -130, 208, 306, -2060, 4796, -5120, -6140, 43320, -113768, 182720, -111768, -395696, 1725172, -3922016, 5614348, -2289912, -14957416, 56700032, -121684568, 164735504, -47657969, -491084768, 1740799985, -3598600386, 4619098604

Offset: 0

Jonas Wallgren

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A051163, A051164, A051166.

a:= proc(n) option remember; add(`if`(k<2, 1,
      a(iquo(k, 2)))*(-1)^(n-k)*binomial(n, k), k=0..n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jul 08 2015

a[n_] := a[n] = Sum[If[k<2, 1, a[Quotient[k, 2]]]*(-1)^(n-k)*Binomial[n, k], {k, 0, n}];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Jun 04 2018, from Maple *)

More terms from Vladeta Jovovic, Jul 26 2002

A051166 Sequence is defined by property that binomial transform of (a0,a1,a2,a3,...) = (a0,a0,a0,a1,a1,a1,a2,a2,a2,a3,a3,a3,...).

Original entry on oeis.org

1, 0, 0, -1, 3, -6, 10, -15, 21, -29, 45, -90, 224, -609, 1677, -4559, 12135, -31542, 80086, -199035, 485469, -1165105, 2757369, -6446778, 14913052, -34175805, 77672325, -175228740, 392711166, -874901088, 1938704130, -4275110880, 9385473510, -20521355211, 44704157499, -97055415324

Offset: 0

Jonas Wallgren

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A051163, A051164, A051165.

a:= proc(n) option remember; add(`if`(k<3, 1,
      a(iquo(k, 3)))*(-1)^(n-k)*binomial(n, k), k=0..n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jul 08 2015

a[n_] := a[n] = Sum[If[k<3, 1, a[Quotient[k, 3]]]*(-1)^(n-k)*Binomial[n, k], {k, 0, n}];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Apr 04 2017, after Alois P. Heinz *)

More terms from Vladeta Jovovic, Jul 26 2002

A137256 Binomial transform of 2^n, 2^n, 2^n.

Original entry on oeis.org

1, 2, 4, 9, 21, 48, 108, 243, 549, 1242, 2808, 6345, 14337, 32400, 73224, 165483, 373977, 845154, 1909980, 4316409, 9754749, 22044960, 49819860, 112588947, 254442141, 575019162, 1299497904, 2936762649, 6636851721, 14998760928

Offset: 0

Paul Curtz, Mar 11 2008

nonn

Sequence is identical to half its third differences.

Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,3).

Cf. A051164, A052103.

I:=[1,2,4]; [n le 3 select I[n] else 3*(Self(n-1) -Self(n-2) +Self(n-3)): n in [1..30]]; // G. C. Greubel, Apr 10 2021

m:=30; S:=series( (1-x+x^2)/(1-3*x+3*x^2-3*x^3), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 10 2021

LinearRecurrence[{3, -3, 3},{1, 2, 4},30] (* Ray Chandler, Sep 23 2015 *)

def A137256_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x+x^2)/(1-3*x+3*x^2-3*x^3) ).list()
A137256_list(30) # G. C. Greubel, Apr 10 2021

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3).

O.g.f.: (1 -x +x^2)/(1 -3*x +3*x^2 -3*x^3). - R. J. Mathar, Apr 02 2008

More terms from R. J. Mathar, Apr 02 2008

A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

Original entry on oeis.org

1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, A8, A1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954

G. C. Greubel, Table of n, a(n) for n = 1..350
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly,66 (1959), 375-384.

Cf. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.
Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.
Cf. A059864, A059865.

[n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023

Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)

a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017

def A059864(n): return product(nth_prime(j) -5 for j in range(4,n+1))
[A059864(n) for n in range(1,31)] # G. C. Greubel, Feb 02 2023

cp's OEIS Frontend

A051163 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a1,a1,a2,a2,a3,a3,...).

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A051165 Sequence is defined by property that binomial transform of (a0,a1,a2,a3,...) = (a0,a0,a1,a1,a2,a2,a3,a3,...).

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A051166 Sequence is defined by property that binomial transform of (a0,a1,a2,a3,...) = (a0,a0,a0,a1,a1,a1,a2,a2,a2,a3,a3,a3,...).

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A137256 Binomial transform of 2^n, 2^n, 2^n.

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A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

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