A227357 -id:A227357 - cp's OEIS frontend

A050983 de Bruijn's S(4,n).

1, 14, 786, 61340, 5562130, 549676764, 57440496036, 6242164112184, 698300344311570, 79881547652046140, 9301427008157320036, 1098786921802152516024, 131361675994216221116836, 15863471168011822803270200, 1932252897656224864335299400

Offset: 0

Eric W. Weisstein

a(n) is divisible by (n+1). Prime p divides a(p-1). Prime p>2 divides all a(n) from a((p+1)/2) to a(p-1). - Alexander Adamchuk, Jul 05 2006

N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.

Seiichi Manyama, Table of n, a(n) for n = 0..471
T. Amdeberhan, De Bruijn's sequence is odd iff n = 2m - 1: Part I, MathOverflow 383035.
Eric Weisstein's World of Mathematics, Binomial Sums

Cf. A000984, A006480, A050984, A099601, A177322, A227357, A249332.

Sum[ (-1)^(k+n)Binomial[ 2n, k ]^4, {k, 0, 2n} ]
RecurrenceTable[{a[0] == 1, a[1] == 14, 4 (n + 1) (2 n + 1)^3 (48 n^2 + 162 n + 137) a[n] + (n + 2)^3 (2 n + 3) (48 n^2 + 66 n + 23) a[n + 2] == 2 (4 (n + 1)^2 (2 n + 3)^2 (408 n^2 + 969 n + 431) - (n + 1) (2 n + 3) (69 n + 31) + 57 n + 92) a[n + 1]}, a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 26 2016 *)

a(n)=sum(k=0,2*n,(-1)^(k+n)*binomial(2*n,k)^4) \\ Charles R Greathouse IV, Dec 28 2011

a(n) = Sum_{k=-n..+n} (-1)^k*C(2*n,n+k)^4. - Benoit Cloitre, Mar 02 2005
a(n) = (-1)^n * HypergeometricPFQ[ {-2n, -2n, -2n, -2n}, {1, 1, 1}, -1]. - Alexander Adamchuk, Jul 05 2006
E.g.f.: Sum(n>=0,I^n*x^n/n!^4) * Sum(n>=0,(-I)^n*x^n/n!^4) = Sum(n>=0,a(n)*x^(2*n)/n!^4) where I^2=-1. - Paul D. Hanna, Dec 21 2011
a(n) ~ 0.125 k^(8n+3)/(Pi*n)^(3/2) where k = 2 cos(Pi/8) = A179260. This formula is due to de Bruijn 1958. - Charles R Greathouse IV, Dec 28 2011
Recurrence: a(0) = 1, a(1) = 14, 4 * (n + 1) * (2*n + 1)^3 * (48*n^2 + 162*n + 137) * a(n) + (n + 2)^3 * (2*n + 3) * (48*n^2 + 66*n + 23) * a(n+2) = 2 * (4 * (n + 1)^2 * (2*n + 3)^2 * (408*n^2 + 969*n + 431) - (n + 1) * (2*n + 3) * (69*n + 31) + 57*n + 92) * a(n+1). - Vladimir Reshetnikov, Sep 26 2016
From Peter Bala, Nov 02 2024; (Start)
a(n) = 1/n * Sum_{k = 0..2*n} (-1)^(n+k) * k * binomial(2*n, k)^4 for n >= 1.
a(n) = binomial(2*n, n) * Sum_{k = 0..n} binomial(2*n, n+k)^2 * binomial(2*n+k,k) = binomial(2*n, n) * Sum_{k = 0..n} (-1)^(n+k) * binomial(2*n, n+k) * binomial(2*n+k, k)^2. (End)

A050984 de Bruijn's S(5,n) = Sum_{k = 0..2n} (-1)^(n+k)binomial(2*n, k)^5.

Original entry on oeis.org

1, 30, 5730, 1696800, 613591650, 248832363780, 108702332138400, 50030418256790400, 23933662070438513250, 11795304320307625903500, 5952113838155498195161980, 3061813957188788125283450400, 1600318610176809076206888362400, 847745162264320796366122559544000

Offset: 0

Eric W. Weisstein

Generally (de Bruijn, 1958), S(s,n) is asymptotic to (2*cos(Pi/(2*s)))^(2*n*s+s-1)*2^(2-s)*(Pi*n)^((1-s)/2)*s^(-1/2). - Vaclav Kotesovec, Jul 09 2013

Andrews (1988) on page 162 states "If, however, we resort to the theory of hypergeometric series, we find that, for example, S(5,n) = - 5F_4[-2n,-2n,-2n,-2n,-2n 1,1,1,1 ; 1]". - _Michael Somos, Jul 24 2013

			1 + 30*x + 5730*x^2 + 1696800*x^3 + 613591650*x^4 + ...

G. E. Andrews "Application of SCRATCHPAD to problems in special functions and combinatorics" Trends in Computer Algebra, R. Janssen, ed., Springer Lecture Notes in Comp.Sci., No. 296, pp. 159-166 (1988)
N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Binomial Sums

Cf. A000984, A006480, A050983, A227357.

Sum[ (-1)^(k+n)Binomial[ 2n, k ]^5, {k, 0, 2n} ]
a[ n_] := If[ n < 0, 0, (-1)^n HypergeometricPFQ[-2 n {1, 1, 1, 1, 1}, {1, 1, 1, 1}, 1]] (* Michael Somos, Jul 24 2013 *)

a(n)=sum(k=0,2*n,(-1)^(k+n)*binomial(2*n,k)^5) \\ Charles R Greathouse IV, Dec 21 2011

E.g.f.: Sum(n>=0,I^n*x^n/n!^5) * Sum(n>=0,(-I)^n*x^n/n!^5) = Sum(n>=0,a(n)*x^(2*n)/n!^5) where I^2=-1. - Paul D. Hanna, Dec 21 2011
a(n) ~ (5+sqrt(5))^(5*n+2)/(sqrt(5)*Pi^2*n^2*2^(5*(n+1))). - Vaclav Kotesovec, Jul 09 2013
Recurrence: n^4*(2*n - 1)^2*(220*n^3 - 858*n^2 + 1119*n - 488)*a(n) = 5*(110000*n^9 - 759000*n^8 + 2252400*n^7 - 3766690*n^6 + 3908325*n^5 - 2609510*n^4 + 1122418*n^3 - 300699*n^2 + 45738*n - 3024)*a(n-1) - 5*(2*n - 3)^2*(5*n - 8)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(220*n^3 - 198*n^2 + 63*n - 7)*a(n-2). - Vaclav Kotesovec, Sep 27 2016
For n >= 1, a(n) = 2 * Sum_{k = 0..2*n-1} (-1)^(n+k) * binomial(2*n, k)^4 * binomial(2*n-1, k) = (1/n) * Sum_{k = 0..2*n} (-1)^(n+k) * k * binomial(2*n, k)^5. - Peter Bala, Oct 31 2024

A350594 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{j=0..2n} (-1)^(n+j) binomial(2*n,j)^k.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 2, 0, -1, 1, 6, 6, 0, 1, 1, 14, 90, 20, 0, -1, 1, 30, 786, 1680, 70, 0, 1, 1, 62, 5730, 61340, 34650, 252, 0, -1, 1, 126, 38466, 1696800, 5562130, 756756, 924, 0, 1, 1, 254, 247170, 41312060, 613591650, 549676764, 17153136, 3432, 0, -1

Offset: 0

Seiichi Manyama, Jan 08 2022

			Square array begins:
   1, 1,   1,      1,         1,            1, ...
  -1, 0,   2,      6,        14,           30, ...
   1, 0,   6,     90,       786,         5730, ...
  -1, 0,  20,   1680,     61340,      1696800, ...
   1, 0,  70,  34650,   5562130,    613591650, ...
  -1, 0, 252, 756756, 549676764, 248832363780, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Columns k=0..6 give A033999, A000007, A000984, A006480, A050983, A050984, A227357.
Rows n=0..1 give A000012, A000918.
Main diagonal gives A350595.
Cf. A257050, A309010.

T(n, k) = sum(j=0, 2*n, (-1)^(n+j)*binomial(2*n, j)^k);

A257050 Array a(m,n) (m>0, n>=0) of quotient of de Bruijn alternating sums of m-th powers of binomial coefficients, listed by ascending antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 15, 1, 0, 1, 15, 131, 84, 1, 0, 1, 31, 955, 3067, 495, 1, 0, 1, 63, 6411, 84840, 79459, 3003, 1, 0, 1, 127, 41195, 2065603, 8765595, 2181257, 18564, 1, 0, 1, 255, 258091, 46942056, 813289963, 987430015, 62165039, 116280, 1, 0

Offset: 1

Jean-François Alcover, Apr 15 2015

			With S(s,n) = de Bruijn sum, array begins:
1,  0,   0,     0,       0,         0,            0, ...
1,  1,   1,     1,       1,         1,            1, ...
1,  3,  15,    84,     495,      3003,        18564, ... = A005809 = S(3,n)/S(2,n)
1,  7, 131,  3067,   79459,   2181257,     62165039, ... = A099601 = S(4,n)/S(2,n)
1, 15, 955, 84840, 8765595, 987430015, 117643216600, ... = S(5,n)/S(2,n)
...
Second column is A000225 (Mersenne numbers).

Neil J. Calkin, Factors of sums of powers of binomial coefficients
Victor J. W. Guo, Frédéric Jouhet and Jiang Zeng, Factors of alternating sums of products of binomial and q-binomial coefficients, arXiv: math.NT/0511635 (2005-2007)
Eric Weisstein's MathWorld, Binomial Sums

Cf. A005809, A006480, A050983, A050984, A099601, A227357.

a[m_, n_] := Sum[(-1)^k*Binomial[2*n, n+k]^m, {k, -n, n}]/Binomial[2*n, n]; Table[a[m-n, n], {m, 1, 10}, {n, 0, m-1}] // Flatten

a(m, n) = (Sum_{k=-n..n} (-1)^k*binomial(2*n, n+k)^m)/binomial(2*n, n).

cp's OEIS Frontend

A050983 de Bruijn's S(4,n).

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A050984 de Bruijn's S(5,n) = Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n, k)^5.

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A350594 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{j=0..2*n} (-1)^(n+j) * binomial(2*n,j)^k.

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Author

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Programs

PARI

A257050 Array a(m,n) (m>0, n>=0) of quotient of de Bruijn alternating sums of m-th powers of binomial coefficients, listed by ascending antidiagonals.

Views

Author

Keywords

Examples

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Programs

Mathematica

Formula

A050984 de Bruijn's S(5,n) = Sum_{k = 0..2n} (-1)^(n+k)binomial(2*n, k)^5.

A350594 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{j=0..2n} (-1)^(n+j) binomial(2*n,j)^k.