A051338 -id:A051338 - cp's OEIS frontend

A051524 Second unsigned column of triangle A051338.

0, 1, 13, 146, 1650, 19524, 245004, 3272688, 46536624, 703404576, 11277554400, 191338156800, 3427105248000, 64651956364800, 1281740285145600, 26648514872985600, 579892995734169600, 13183403757582643200

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=6) ~ exp(-x)/x^2*(1 - 13/x + 146/x^2 - 1650/x^3 + 19524/x^4 - 245004/x^5 + 3272688/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009

Mitrinovic, D. S. and Mitrinovic, R. S.: see reference given for triangle A051338.

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A001725 (first unsigned column).

Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564. - Gary Detlefs, Jan 04 2011

f[k_] := k + 5; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]
(* Clark Kimberling, Dec 29 2011 *)

a(n) = A051338(n, 1)*(-1)^(n-1);
E.g.f.: -log(1-x)/(1-x)^6.
For n>=1, a(n) = n!*Sum_{k=0..n-1} (-1)^k*binomial(-6,k)/(n-k). - Milan Janjic, Dec 14 2008
a(n) = n!*[5]h(n), where [k]h(n) denotes the k-th successive summation of h(n) from 0 to n. - Gary Detlefs, Jan 04 2011
Conjecture: a(n) +(-2*n-9)*a(n-1) +(n+4)^2*a(n-2)=0. - R. J. Mathar, Aug 04 2013

A051525 Third unsigned column of triangle A051338.

Original entry on oeis.org

0, 0, 1, 21, 335, 5000, 74524, 1139292, 18083484, 299705400, 5198985576, 94461323616, 1797180658272, 35776357096896, 744402741205824, 16169795109262080, 366214212167489280, 8636605663418933760

Offset: 0

Wolfdieter Lang

From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=3,n=6) ~ exp(-x)/x^3*(1 - 21/x + 335/x^2 - 5000/x^3 + 74524/x^4 - 1139292/x^5 + ...) leads to the sequence given above. See A163931 and A163932 for more information.
(End)

Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051338.

Cf. A001725 (m=0), A051524 (m=1) unsigned columns.

a(n) = A051338(n, 2)*(-1)^n; e.g.f.: (log(1-x))^2/(2*(1-x)^6).

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = |f(n,2,6)|, for n>=1. - Milan Janjic, Dec 21 2008

A001720 a(n) = n!/24.

Original entry on oeis.org

1, 5, 30, 210, 1680, 15120, 151200, 1663200, 19958400, 259459200, 3632428800, 54486432000, 871782912000, 14820309504000, 266765571072000, 5068545850368000, 101370917007360000, 2128789257154560000, 46833363657400320000, 1077167364120207360000

Offset: 4

N. J. A. Sloane

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=5) ~ exp(-x)/x*(1 - 5/x + 30/x^2 - 210/x^3 + 1680/x^4 - 15120/x^5 + 151200/x^6 - 1663200/x^7 + ...) leads to this sequence. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 4..300
S. Cerdá, A simple scheme to find the solutions to Brocard-Ramanujan Diophantine Equation n! + 1 = m^2, arXiv:1504.06694 [math.NT], 2015.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 264.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A000142, A049353, A049459, A051338, A245334.
Cf. A001710, A001715, A007696, A094638.
Cf. A094587, A130534, A138533, A163931, A173333, A213936.

a001720 = (flip div 24) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/24: n in [4..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_]:=n!/24; (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
Range[4,30]!/24 (* Harvey P. Dale, Jul 27 2021 *)

a(n)=n!/24 \\ Charles R Greathouse IV, Jan 12 2012

a(n)= A049353(n-3, 1) (first column of triangle).
E.g.f. if offset 0: 1/(1-x)^5.
a(n) = A173333(n,4). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n,n-4) / 5. - Reinhard Zumkeller, Aug 31 2014
G(x) = (1 - (1 + x)^(-4)) / 4 = x - 5 x^2/2! + 30 x^3/3! - ..., an e.g.f. for this signed sequence (for n!/4!), is the compositional inverse of H(x) = (1 - 4*x)^(-1/4) - 1 = x + 5 x^2/2! + 45 x^3/3! + ..., an e.g.f. for A007696. Cf. A094638, A001710 (for n!/2!), and A001715 (for n!/3!). Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019
E.g.f.: x^4 / (4! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=4} 1/a(n) = 24*e - 64.
Sum_{n>=4} (-1)^n/a(n) = 24/e - 8. (End)

A001725 a(n) = n!/5!.

Original entry on oeis.org

1, 6, 42, 336, 3024, 30240, 332640, 3991680, 51891840, 726485760, 10897286400, 174356582400, 2964061900800, 53353114214400, 1013709170073600, 20274183401472000, 425757851430912000, 9366672731480064000, 215433472824041472000, 5170403347776995328000

Offset: 5

N. J. A. Sloane

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=6) ~ exp(-x)/x*(1 - 6/x + 42/x^2 - 336/x^3 + 3024/x^4 - 30240/x^5 + 332640/x^6 - 3991680/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 5..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 265.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. II, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 107-108 1963 1-77.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

a(n)= A049374(n-4), n >= 1 (first column of triangle). Cf. A049460, A051339. a(n)= A051338(n-5, 0)*(-1)^(n-1) (first unsigned column of triangle).

Cf. A245334, A000142.

a001725 = (flip div 120) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/120: n in [5..25]]; // Vincenzo Librandi, Jul 20 2011

lst={};Do[AppendTo[lst, n!/5! ], {n, 5, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
Range[5,30]!/120 (* Harvey P. Dale, Dec 20 2014 *)

a(n)=n!/120 \\ Charles R Greathouse IV, Jul 19 2011

E.g.f. if offset 0: 1/(1-x)^6.
a(n) = A173333(n,5). - Reinhard Zumkeller, Feb 19 2010
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+6)/(x*(k+6) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: W(0)/(40*x^2) -1/(20*x^2) -1/(5*x) , where W(k) = 1 + 1/( 1 - x*(k+4)/( x*(k+4) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013
a(n) = A245334(n,n-5) / 6. - Reinhard Zumkeller, Aug 31 2014
E.g.f.: x^5 / (5! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=5} 1/a(n) = 120*e - 325.
Sum_{n>=5} (-1)^(n+1)/a(n) = 45 - 120/e. (End)

More terms from Harvey P. Dale, Dec 20 2014

A001730 a(n) = n!/6!.

Original entry on oeis.org

1, 7, 56, 504, 5040, 55440, 665280, 8648640, 121080960, 1816214400, 29059430400, 494010316800, 8892185702400, 168951528345600, 3379030566912000, 70959641905152000, 1561112121913344000, 35905578804006912000, 861733891296165888000, 21543347282404147200000

Offset: 6

N. J. A. Sloane

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=7) ~ exp(-x)/x*(1 - 7/x + 56/x^2 - 504/x^3 + 5040/x^4 - 55440/x^5 + 665280/x^6 - 8648640/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 6..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 266.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. II, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 107-108 1963 1-77.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A051338, A051339, A051379.

Cf. A130534, A163931, A173333, A245334, A000142.

a001730 = (flip div 720) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/720: n in [6..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_]:=n!/6!; Array[a,4!,6] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2009 *)

a(n)=n!/720 \\ Charles R Greathouse IV, Jan 12 2012

a(n)= A051339(n-6, 0)*(-1)^n (first unsigned column of triangle).
E.g.f.: x^6/(6!*(1-x)). [corrected by Alois P. Heinz, Jul 09 2021]
a(n) = A173333(n,6). - Reinhard Zumkeller, Feb 19 2010
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+7)/(x*(k+7) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
a(n) = A245334(n,n-6) / 7. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=6} 1/a(n) = 720*e - 1956.
Sum_{n>=6} (-1)^n/a(n) = 720/e - 264. (End)

A105954 Array read by descending antidiagonals: A(n, k) = (n + 1)! * H(k, n + 1), where H(n, k) is a higher-order harmonic number, H(0, k) = 1/k and H(n, k) = Sum_{j=1..k} H(n-1, j), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 11, 6, 1, 7, 26, 50, 24, 1, 9, 47, 154, 274, 120, 1, 11, 74, 342, 1044, 1764, 720, 1, 13, 107, 638, 2754, 8028, 13068, 5040, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320, 1, 17, 191, 1650, 11274, 60216, 241128, 663696, 1026576, 362880

Offset: 0

Leroy Quet, Jun 26 2005

Antidiagonal sums are A093345 (n! * (1 + Sum_{i=1..n}((1/i)*Sum_{j=0..i-1} 1/j!))). - Gerald McGarvey, Aug 27 2005
A recasting of A093905 and A067176. - R. J. Mathar, Mar 01 2009
The triangular array of this sequence is the reversal of A165675 which is related to the asymptotic expansion of the higher order exponential integral E(x,m=2,n); see also A165674. - Johannes W. Meijer, Oct 16 2009

			A(2, 2) = (1 + (1 + 1/2) + (1 + 1/2 + 1/3))*6 = 26.
Array A(n, k) begins:
  [n\k]  0       1       2        3        4        5          6
  -------------------------------------------------------------------
  [0]    1,      1,      1,       1,       1,       1,         1, ...
  [1]    1,      3,      5,       7,       9,       11,       13, ...
  [2]    2,     11,     26,      47,      74,      107,      146, ...
  [3]    6,     50,    154,     342,     638,     1066,     1650, ...
  [4]   24,    274,   1044,    2754,    5944,    11274,    19524, ...
  [5]  120,   1764,   8028,   24552,   60216,   127860,   245004, ...
  [6]  720,  13068,  69264,  241128,  662640,  1557660,  3272688, ...
  [7] 5040, 109584, 663696, 2592720, 7893840, 20355120, 46536624, ...

G. C. Greubel, Table of n, a(n) for the first 27 rows, flattened
Arthur T. Benjamin, David Gaebler and Robert Gaebler, A Combinatorial Approach to Hyperharmonic Numbers, INTEGERS, Electronic Journal of Combinatorial Number Theory, Volum 3, #A15, 2003.

Column 0 = A000142 (factorial numbers).
Column 1 = A000254 (Stirling numbers of first kind s(n, 2)) starting at n=1.
Column 2 = A001705 (Generalized Stirling numbers: a(n) = n!*Sum_{k=0..n-1}(k+1)/(n-k)), starting at n=1.
Column 3 = A001711 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*3^k*stirling1(n+1, k+1)).
Column 4 = A001716 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*4^k*stirling1(n+1, k+1)).
Column 5 = A001721 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*binomial(k+1, 1)*5^k*stirling1(n+1, k+1)).
Column 6 = A051524 (2nd unsigned column of A051338) starting at n=1.
Column 7 = A051545 (2nd unsigned column of A051339) starting at n=1.
Column 8 = A051560 (2nd unsigned column of A051379) starting at n=1.
Column 9 = A051562 (2nd unsigned column of A051380) starting at n=1.
Column 10= A051564 (2nd unsigned column of A051523) starting at n=1.
2nd row is A005408 (2n - 1, starting at n=1).
3rd row is A080663 (3n^2 - 1, starting at n=1).
Main diagonal gives A384024.
Cf. A000254, A165674 and A165675, A028421 and A126671.

H := proc(n, k) option remember; if n = 0 then 1/k else add(H(n - 1, j), j = 1..k) fi end: A := (n, k) -> (n + 1)!*H(k, n + 1):
# Alternative with standard harmonic number:
A := (n, k) -> if k = 0 then n! else (harmonic(n + k) - harmonic(k - 1))*(n + k)! / (k - 1)! fi:
for n from 0 to 7 do seq(A(n, k), k = 0..6) od;
# Alternative with hypergeometric formula:
A := (n, k) -> (n+1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k+1], 1):
seq(print(seq(simplify(A(n, k)), k = 0..6)), n=0..7); # Peter Luschny, Jul 01 2022

H[0, m_] := 1/m; H[n_, m_] := Sum[H[n - 1, k], {k, m}]; a[n_, m_] := m!H[n, m]; Flatten[ Table[ a[i, n - i], {n, 10}, {i, n - 1, 0, -1}]]
Table[ a[n, m], {m, 8}, {n, 0, m + 1}] // TableForm (* to view the table *)
(* Robert G. Wilson v, Jun 27 2005 *)

a(n, k) = polcoef(prod(j=0, n, 1+(j+k)*x), n); \\ Seiichi Manyama, May 19 2025

A(n, k) = (Harmonic(n + k) - Harmonic(k - 1))*(n + k)!/(k - 1)! if k > 0, otherwise n!.
From Gerald McGarvey, Aug 27 2005, edited by Peter Luschny, Jul 02 2022: (Start)
E.g.f. for column k: -log(1 - x)/(x*(1 - x)^k).
Row 3 is r(n) = 4*n^3 + 18*n^2 + 22*n + 6.
Row 4 is r(n) = 5*n^4 + 40*n^3 + 105*n^2 + 100*n + 24.
Row 5 is r(n) = 6*n^5 + 75*n^4 + 340*n^3 + 675*n^2 + 548*n + 120.
Row 6 is r(n) = 7*n^6 + 126*n^5 + 875*n^4 + 2940*n^3 + 4872*n^2 + 3528*n + 720.
Row 7 is r(n) = 8*n^7 + 196*n^6 + 1932*n^5 + 9800*n^4 + 27076*n^3 + 39396*n^2 + 26136*n + 5040.
The sum of the polynomial coefficients for the n-th row is |S1(n, 2)|, which are the unsigned Stirling1 numbers which appear in column 1.
A(m, n) = Sum_{k=1..m} n*A094645(m, n)*(n+1)^(k-1). (A094645 is Generalized Stirling number triangle of first kind, e.g.f.: (1-y)^(1-x).) (End)
In Gerard McGarvey's formulas for the row coefficients we find Wiggen's triangle A028421 and their o.g.f.s lead to Wood's polynomials A126671; see A165674. - Johannes W. Meijer, Oct 16 2009
A(n, k) = (n + 1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k + 1], 1). - Peter Luschny, Jul 01 2022
A(n,k) = [x^n] Product_{j=0..n} (1 + (j+k)*x). - Seiichi Manyama, May 19 2025

More terms from Robert G. Wilson v, Jun 27 2005

Edited by Peter Luschny, Jul 02 2022

A051523 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -10, 1, 110, -21, 1, -1320, 362, -33, 1, 17160, -6026, 791, -46, 1, -240240, 101524, -17100, 1435, -60, 1, 3603600, -1763100, 358024, -38625, 2335, -75, 1, -57657600, 31813200, -7491484, 976024, -75985, 3535, -91, 1, 980179200, -598482000, 159168428, -24083892, 2267769, -136080, 5082, -108, 1

Offset: 0

Wolfdieter Lang

a(n,m)= ^10P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(10+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(10*t),exp(t)-1).

			{1}; {-10,1}; {110,-21,1}; {-1320,362,-331}; ... s(2,x)= 110-21*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

The first (m=0) unsigned column sequence is A049398. Row sums (signed triangle): A049389(n)*(-1)^n. Row sums (unsigned triangle): A051431(n).
See also A049444 A049458 A049459 A049460 A051338 A051339 A051379 A051380
Cf. A000035 A084938.

a051523 n k = a051523_tabl !! n !! k
a051523_row n = a051523_tabl !! n
a051523_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 10)
-- Reinhard Zumkeller, Mar 12 2014

a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^10, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

a(n, m)= a(n-1, m-1) - (n+9)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^10).

Triangle (signed) = [ -10, -1, -11, -2, -12, -3, -13, -14, -4, ...] DELTA A000035; triangle (unsigned) = [10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,10), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

A087748 Triangle formed by reading triangle of Stirling numbers of the first kind (A048994) mod 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 0

Philippe Deléham, Oct 02 2003

			Triangle begins:
1,
0, 1,
0, 1, 1,
0, 0, 1, 1,
0, 0, 1, 0, 1,
0, 0, 0, 1, 0, 1,
0, 0, 0, 1, 1, 1, 1,
0, 0, 0, 0, 1, 1, 1, 1,
0, 0, 0, 0, 1, 0, 0, 0, 1,
0, 0, 0, 0, 0, 1, 0, 0, 0, 1,
0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1,
0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1,
...

Brand, Neal; Das, Sajal; Jacob, Tom. The number of nonzero entries in recursively defined tables modulo primes. Proceedings of the Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1990). Congr. Numer. 78 (1990), 47--59. MR1140469 (92h:05004). - From N. J. A. Sloane, Jun 03 2012

Bill Gosper, Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7

Cf. A008275, A008276, A048994, A087755.

Also parity of triangles A049444, A049459, A051338, A051379, A051523.

T(n, k) = A087755(n, k) = A048994(n, k) mod 2 = A047999([n/2], k-[(n+1)/2]) = T(n-2, k-2) XOR T(n-2, k-1) with T(0, 0) = T(1, 1) = 1 and T(1, 0) = 0; T(2n, k) = T(2n-1, k-1) XOR T(2n-1, k); T(2n+1, k) = T(2n, k-1). - Henry Bottomley, Dec 01 2003

Edited and extended by Henry Bottomley, Dec 01 2003

A087745 Numbers A001317 repeated.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 15, 15, 17, 17, 51, 51, 85, 85, 255, 255, 257, 257, 771, 771, 1285, 1285, 3855, 3855, 4369, 4369, 13107, 13107, 21845, 21845, 65535, 65535, 65537, 65537, 196611, 196611, 327685, 327685, 983055, 983055, 1114129, 1114129

Offset: 0

Philippe Deléham, Oct 02 2003

Triangles A049444, A049459, A051338, A051379, A051523 (Mitrinovic's triangles) mod 2 converted to decimal.

Sequence [1, 3, 5, 15, 17, 51, 85, 255, 257, ...] = A001317.

Cf. A001317, A049444, A049459, A051338, A051378, A051523.

def A087745(n): return sum((bool(~(m:=n>>1)&m-k)^1)<>1)+1)) # Chai Wah Wu, May 02 2023

Definition corrected and edited by Omar E. Pol, Dec 24 2008

Showing 1-9 of 9 results.

cp's OEIS Frontend

A051524 Second unsigned column of triangle A051338.

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A051525 Third unsigned column of triangle A051338.

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A001720 a(n) = n!/24.

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A001725 a(n) = n!/5!.

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A001730 a(n) = n!/6!.

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A105954 Array read by descending antidiagonals: A(n, k) = (n + 1)! * H(k, n + 1), where H(n, k) is a higher-order harmonic number, H(0, k) = 1/k and H(n, k) = Sum_{j=1..k} H(n-1, j), for 0 <= k <= n.

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A051523 Generalized Stirling number triangle of first kind.

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