A051163 -id:A051163 - cp's OEIS frontend

A100936 Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)1, C(n,1)2,..., C(n,k)A051163(k), ..., C(n,n)A051163(n)] and where A051162 equals the antidiagonal sums.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 28, 28, 9, 1, 1, 11, 47, 76, 47, 11, 1, 1, 13, 71, 163, 163, 71, 13, 1, 1, 15, 100, 301, 435, 301, 100, 15, 1, 1, 17, 134, 502, 971, 971, 502, 134, 17, 1, 1, 19, 173, 778, 1909, 2577, 1909, 778, 173, 19, 1, 1, 21, 217, 1141

Offset: 0

Paul D. Hanna, Nov 23 2004

Antidiagonal sums form A051163. Main diagonal is A100937. Different from A086620.

			Rows begin:
[1,1,1,1,1,1,1,1,1,...],
[1,3,5,7,9,11,13,15,17,...],
[1,5,14,28,47,71,100,134,...],
[1,7,28,76,163,301,502,778,...],
[1,9,47,163,435,971,1909,3417,...],
[1,11,71,301,971,2577,5917,12167,...],
[1,13,100,502,1909,5917,15678,36744,...],
[1,15,134,778,3417,12167,36744,97272,...],...
Antidiagonal sums form A051163: [1,2,5,12,30,76,194,496,1269,3250,8337,...].
The inverse binomial transform of the rows form the respective rows of the triangle B:
[1*1],
[1*1,1*2],
[1*1,2*2,1*5],
[1*1,3*2,3*5,1*12],
[1*1,4*2,6*5,4*12,1*30],...
where B(n,k) = binomial(n,k)*A051163(k).

Cf. A051163, A100937.

T(n,k)=if(n==0 || k==0,1, sum(j=0,n,binomial(k,j)*binomial(n,j)*sum(i=0,j,T(j-i,i)));)

T(n, k) = Sum_{j=0..n} C(k, j)*C(n, j)*A051162(j), with T(0, 0) = 1 and where Sum_{i=0..n} T(n-i, i) = A051162(n).

A152198 Triangle read by rows, A007318 rows repeated.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1

Offset: 0

Gary W. Adamson, Nov 28 2008

Eigensequence of the triangle = A051163: (1, 2, 5, 12, 30, 76,...)
Another version of A152815. - Philippe Deléham, Dec 13 2008
Row sums : A016116(n); Diagonal sums: A000931(n+5). - Philippe Deléham, Dec 13 2008
Triangle, with zeros omitted, given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 16 2012
Sums along rising diagonals are A134816. - John Molokach, Jul 09 2013

			The triangle starts
1;
1;
1, 1;
1, 1;
1, 2, 1;
1, 2, 1;
1, 3, 3, 1;
1, 3, 3, 1;
1, 4, 6, 4, 1;
1, 4, 6, 4, 1;
1, 5, 10, 10, 5, 1;
1, 5, 10, 10, 5, 1;
...
Triangle (1,0,-1,0,0,...) DELTA (0,1,-1,0,0,...) begins:
1
1, 0
1, 1, 0
1, 1, 0, 0
1, 2, 1, 0, 0
1, 2, 1, 0, 0, 0
1, 3, 3, 1, 0, 0, 0
1, 3, 3, 1, 0, 0, 0, 0
1, 4, 6, 4, 1, 0, 0, 0, 0
1, 4, 6, 4, 1, 0, 0, 0, 0, 0
1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 0...

Cf. A007318, A133156, A051163.

t[n_, k_] := Binomial[ Floor[n/2], k]; Table[t[n, k], {n, 0, 17}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Sep 13 2012 *)

Triangle read by rows, Pascal's triangle rows repeated.
Equals inverse binomial transform of A133156 unsigned.
G.f. : (1+x)/(1-(1+y)*x^2). - Philippe Deléham, Jan 16 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057077(n), A019590(n+1), A000012(n), A016116(n), A108411(n), A074872(n+1) for x = -2, -1, 0, 1, 2, 4 respectively. - Philippe Deléham, Jan 16 2012
T(n,k) = A065941(n-k, n-2*k) = abs(A108299(n-k, n-2*k)). - Johannes W. Meijer, Sep 05 2013

More terms from Philippe Deléham, Dec 14 2008

A027826 Inverse binomial transform of a_0 = 1, a_1, a_2, etc. is a_0, 0, a_1, 0, a_2, 0, etc.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 50, 120, 290, 706, 1732, 4280, 10644, 26612, 66824, 168384, 425481, 1077529, 2733746, 6945812, 17669149, 44994345, 114682042, 292544200, 746831570, 1907983346, 4877966628, 12479883736, 31951158024, 81858610968, 209865391600, 538408691456

Offset: 0

Allan C. Wechsler

The self-convolution equals A051163. - Paul D. Hanna, Nov 23 2004

Equals row sums of triangle A152193. - Gary W. Adamson, Nov 28 2008

			Array of successive differences (col. 1 is the inverse binomial transform):
1, 1,  2,  4,  9, 21, 50, ...
0, 1,  2,  5, 12, 29, 70, ...
1, 1,  3,  7, 17, 41, ...
0, 2,  4, 10, 24, 59, ...
2, 2,  6, 14, 35, 87, ...
0, 4,  8, 21, 52, ...
4, 4, 13, 31, 79, ...
0, 9, 18, 48, ...
9, 9, 30, ...
...

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

Cf. A051163.

Cf. A152193. - Gary W. Adamson, Nov 28 2008

a:= proc(n) option remember; add(`if`(k=0, 1,
      `if`(k::odd, 0, a(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] = Sum[If[k == 0, 1, If[OddQ[k], 0, a[k/2]]]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 20 2017, translated from Maple *)

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,n))

a=List();for(n=1,100,listput(a,sum(i=1,n\2,a[i]*binomial(n,2*i),1))) \\ M. F. Hasler, Aug 19 2015

G.f. A(x) satisfies A(x^2) = A(x/(1+x))/(1+x) and A(x) = A(x^2/(1-x)^2)/(1-x).
The recursive formula A[n+1] = A[n](x^2/(1-x)^2)/(1-x), A[0]=1, yields exactly 2^n terms after n iterations: A(x) - A[n](x) = x^(2^n) + (2^n+1)*x^(2^n+1) + O(x^(2^n+2)). For example, A[4] = (1-x)^3*(1-2*x-x^2)/((1-2*x)(1-4*x+4*x^2-2*x^4)) = A(x) - x^16 - 17*x^17 + O(x^18). - M. F. Hasler, Aug 19 2015
E.g.f.: exp(x) * Sum_{n>=0} a(n) * x^(2*n) / (2*n)!. - Ilya Gutkovskiy, Feb 26 2022
The expansion of exp(Sum_{n >= 1} a(n)*(2*x)^n/n!) = 1 + 2*x + 6*x^2 + 20*x^3 + 74*x^4 + 292*x^5 + 1204*x^6 + ... appears to have integer coefficients. Equivalently, the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for positive integers k and n and all primes p >= 3. - Peter Bala, Jan 11 2023

Incorrect g.f. and formulas removed by R. J. Mathar, Oct 02 2012

Incorrect g.f.s removed by Peter Bala, Jul 07 2015

A051164 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a0,a1,a1,a1,a2,a2,a2,a3,a3,a3,...).

Original entry on oeis.org

1, 2, 4, 9, 21, 48, 108, 243, 549, 1243, 2819, 6412, 14640, 33549, 77121, 177747, 410565, 949992, 2200978, 5103623, 11839783, 27471189, 63734823, 147831594, 342767586, 794413545, 1840338975, 4261443374, 9863627962, 22822212734, 52789053456, 122073285984

Offset: 0

Jonas Wallgren

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

Cf. A051163, A051165, A051166.

a:= proc(n) option remember; add(`if`(k<3, 1,
      a(iquo(k, 3)))*binomial(n, k), k=0..n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 08 2015

a[n_] := a[n] = Sum[If[k<3, 1, a[Quotient[k, 3]]]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017, after Alois P. Heinz *)

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

A100937 Main diagonal of symmetric square array A100936.

Original entry on oeis.org

1, 3, 14, 76, 435, 2577, 15678, 97272, 612126, 3891890, 24933292, 160663328, 1040074684, 6759228932, 44075916696, 288289595968, 1890894150707, 12434303045721, 81960791460442, 541428229233012, 3583843659376257

Offset: 0

Paul D. Hanna, Nov 23 2004

nonn

			a(3) = 76 = 1*1 + 3^2*2 + 3^2*5 + 1*12 = Sum_{k=0..3} C(3,k)^2*A051163(k).
a(4) = 435 = 1*1 + 4^2*2 + 6^2*5 + 4^2*12 + 1*30 = Sum_{k=0..4} C(4,k)^2*A051163(k).

Cf. A051163, A100936.

PARI

a(n) = Sum_{k=0..n} C(n, k)^2*A051162(k).

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

A352901 a(0) = 1; a(n) = Sum_{k=0..floor(n/3)} binomial(n+1,3k+1) a(k).

Original entry on oeis.org

1, 2, 3, 6, 15, 36, 80, 172, 369, 796, 1727, 3774, 8322, 18528, 41643, 94460, 216121, 498186, 1155147, 2689626, 6278841, 14676900, 34316598, 80194032, 187195554, 436310190, 1015176726, 2357708258, 5465611759, 12647864454, 29219750157, 67403414568, 155276809533

Offset: 0

Ilya Gutkovskiy, Apr 07 2022

nonn

Self-convolution of A351970.

Cf. A040027, A051163, A351970, A352902.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 1, 3 k + 1] a[k], {k, 0, Floor[n/3]}]; Table[a[n], {n, 0, 32}]
nmax = 32; A[] = 1; Do[A[x] = A[x^3/(1 - x)^3]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = A(x^3/(1 - x)^3) / (1 - x)^2.

E.g.f.: d/dx ( exp(x) * Sum_{n>=0} a(n) * x^(3*n+1) / (3*n+1)! ).

A352902 a(0) = 1; a(n) = Sum_{k=0..floor(n/4)} binomial(n+1,4k+1) a(k).

Original entry on oeis.org

1, 2, 3, 4, 7, 18, 49, 120, 264, 544, 1100, 2256, 4736, 10080, 21456, 45312, 94850, 197412, 410134, 852968, 1778382, 3719364, 7806338, 16451280, 34835184, 74164800, 158854536, 342478432, 743432288, 1625079936, 3576582592, 7922065408, 17648409603

Offset: 0

Ilya Gutkovskiy, Apr 07 2022

nonn

Self-convolution of A351971.

Cf. A040027, A051163, A351971, A352901.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 1, 4 k + 1] a[k], {k, 0, Floor[n/4]}]; Table[a[n], {n, 0, 32}]
nmax = 32; A[] = 1; Do[A[x] = A[x^4/(1 - x)^4]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = A(x^4/(1 - x)^4) / (1 - x)^2.

E.g.f.: d/dx ( exp(x) * Sum_{n>=0} a(n) * x^(4*n+1) / (4*n+1)! ).

cp's OEIS Frontend

A100936 Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)*1, C(n,1)*2,..., C(n,k)*A051163(k), ..., C(n,n)*A051163(n)] and where A051162 equals the antidiagonal sums.

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A152198 Triangle read by rows, A007318 rows repeated.

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A027826 Inverse binomial transform of a_0 = 1, a_1, a_2, etc. is a_0, 0, a_1, 0, a_2, 0, etc.

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A051164 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a0,a1,a1,a1,a2,a2,a2,a3,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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A100937 Main diagonal of symmetric square array A100936.

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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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A100936 Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)1, C(n,1)2,..., C(n,k)A051163(k), ..., C(n,n)A051163(n)] and where A051162 equals the antidiagonal sums.

A352901 a(0) = 1; a(n) = Sum_{k=0..floor(n/3)} binomial(n+1,3k+1) a(k).

A352902 a(0) = 1; a(n) = Sum_{k=0..floor(n/4)} binomial(n+1,4k+1) a(k).