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 1-5 of 5 results.

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

Original entry on oeis.org

1, 2, 0, 3, 0, 1, 4, 0, 4, 0, 5, 0, 10, 0, 1, 6, 0, 20, 0, 6, 0, 7, 0, 35, 0, 21, 0, 1, 8, 0, 56, 0, 56, 0, 8, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 12, 0, 220, 0, 792, 0, 792, 0, 220, 0, 12, 0
Offset: 0

Views

Author

Philippe Deléham, Mar 05 2014

Keywords

Comments

Row sums are powers of 2.

Examples

			Triangle begins:
1;
2, 0;
3, 0, 1;
4, 0, 4, 0;
5, 0, 10, 0, 1;
6, 0, 20, 0, 6, 0;
7, 0, 35, 0, 21, 0, 1;
8, 0, 56, 0, 56, 0, 8, 0;
9, 0, 84, 0, 126, 0, 36, 0, 1;
10, 0, 120, 0, 252, 0, 120, 0, 10, 0; etc.

Links

Crossrefs

Cf. Columns: A000027, A000292, A000389, A000580, A000582, A001288, A010966, A010968, A010970, A010972, A010974, A010976, A010980, A010982, A010984, A010986, A010988, A010990, A010992, A010994, A010996, A010998, A011000, A017713, A017715, A017717, A017719, A017721, A017723, A017725, A017727, A017729, A017731, A017733, A017735, A017737, A017739, A017741, A017743, A017745, A017747, A017749, A017751, A017753, A017755, A017757, A017759, A017761, A017763.
Cf. A095704, A178616.

Programs

  • Mathematica
    Table[Binomial[n + 1, k + 1]*(1 - Mod[k , 2]), {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 22 2017 *)
  • PARI
    T(n,k) = binomial(n+1, k+1)*(1-(k % 2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 23 2017

Formula

G.f.: 1/((1+(y-1)*x)*(1-(y+1)*x)).
T(n,k) = 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A000079(n), A015518(n+1), A003683(n+1), A079773(n+1), A051958(n+1), A080920(n+1), A053455(n), A160958(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A010991 Binomial coefficient C(n,38).

Original entry on oeis.org

1, 39, 780, 10660, 111930, 962598, 7059052, 45379620, 260932815, 1362649145, 6540715896, 29135916264, 121399651100, 476260169700, 1768966344600, 6250347750920, 21094923659355, 68248282427325, 212327989773900, 636983969321700, 1847253511032930, 5189902721473470
Offset: 38

Views

Author

N. J. A. Sloane

Keywords

Links

  • T. D. Noe, Table of n, a(n) for n = 38..1000
  • Index entries for linear recurrences with constant coefficients, signature (39, -741, 9139, -82251, 575757, -3262623, 15380937, -61523748, 211915132, -635745396, 1676056044, -3910797436, 8122425444, -15084504396, 25140840660, -37711260990, 51021117810, -62359143990, 68923264410, -68923264410, 62359143990, -51021117810, 37711260990, -25140840660, 15084504396, -8122425444, 3910797436, -1676056044, 635745396, -211915132, 61523748, -15380937, 3262623, -575757, 82251, -9139, 741, -39, 1).

Crossrefs

Cf. A010988, A010989, A010990, A001787, A242091.

Programs

Formula

G.f.: x^38/(1-x)^39. - Zerinvary Lajos, Dec 19 2008; adapted to offset by Enxhell Luzhnica, Jan 23 2017
From Amiram Eldar, Dec 15 2020: (Start)
Sum_{n>=38} 1/a(n) = 38/37.
Sum_{n>=38} (-1)^n/a(n) = A001787(38)*log(2) - A242091(38)/37! = 5222680231936*log(2) - 31812289115113208816827133/8787716212275 = 0.9755552351... (End)

A010992 Binomial coefficient C(n,39).

Original entry on oeis.org

1, 40, 820, 11480, 123410, 1086008, 8145060, 53524680, 314457495, 1677106640, 8217822536, 37353738800, 158753389900, 635013559600, 2403979904200, 8654327655120, 29749251314475, 97997533741800, 310325523515700, 947309492837400, 2794563003870330
Offset: 39

Views

Author

N. J. A. Sloane

Keywords

Links

  • T. D. Noe, Table of n, a(n) for n = 39..1000
  • Index entries for linear recurrences with constant coefficients, signature (40, -780, 9880, -91390, 658008, -3838380, 18643560, -76904685, 273438880, -847660528, 2311801440, -5586853480, 12033222880, -23206929840, 40225345056, -62852101650, 88732378800, -113380261800, 131282408400, -137846528820, 131282408400, -113380261800, 88732378800, -62852101650, 40225345056, -23206929840, 12033222880, -5586853480, 2311801440, -847660528, 273438880, -76904685, 18643560, -3838380, 658008, -91390, 9880, -780, 40, -1).

Crossrefs

Cf. A010990, A010991, A001787, A242091.

Programs

Formula

G.f.: x^39/(1-x)^40. - Zerinvary Lajos, Dec 19 2008
From Amiram Eldar, Dec 15 2020: (Start)
Sum_{n>=39} 1/a(n) = 39/38.
Sum_{n>=39} (-1)^(n+1)/a(n) = A001787(39)*log(2) - A242091(39)/38! = 10720238370816*log(2) - 63624578230235205349866541/8562390155550 = 0.9761396932... (End)

A010994 a(n) = binomial coefficient C(n,41).

Original entry on oeis.org

1, 42, 903, 13244, 148995, 1370754, 10737573, 73629072, 450978066, 2505433700, 12777711870, 60403728840, 266783135710, 1108176102180, 4353548972850, 16253249498640, 57902201338905, 197548686920970, 647520696018735, 2044802197953900, 6236646703759395
Offset: 41

Views

Author

N. J. A. Sloane

Keywords

Links

  • T. D. Noe, Table of n, a(n) for n = 41..1000
  • Index entries for linear recurrences with constant coefficients, signature (42, -861, 11480, -111930, 850668, -5245786, 26978328, -118030185, 445891810, -1471442973, 4280561376, -11058116888, 25518731280, -52860229080, 98672427616, -166509721602, 254661927156, -353697121050, 446775310800, -513791607420, 538257874440, -513791607420, 446775310800, -353697121050, 254661927156, -166509721602, 98672427616, -52860229080, 25518731280, -11058116888, 4280561376, -1471442973, 445891810, -118030185, 26978328, -5245786, 850668, -111930, 11480, -861, 42, -1).

Crossrefs

Cf. A010990, A010991, A010992, A001787, A242091.

Programs

Formula

G.f.: x^41/(1-x)^42. - Zerinvary Lajos, Dec 20 2008
From Amiram Eldar, Dec 15 2020: (Start)
Sum_{n>=41} 1/a(n) = 41/40.
Sum_{n>=41} (-1)^(n+1)/a(n) = A001787(41)*log(2) - A242091(41)/40! = 45079976738816*log(2) - 41737723319038472299669343741/1335732864265800 = 0.9772284535... (End)

A095704 Triangle read by rows giving coefficients of the trigonometric expansion of sin(n*x).

Original entry on oeis.org

1, 2, 0, 3, 0, -1, 4, 0, -4, 0, 5, 0, -10, 0, 1, 6, 0, -20, 0, 6, 0, 7, 0, -35, 0, 21, 0, -1, 8, 0, -56, 0, 56, 0, -8, 0, 9, 0, -84, 0, 126, 0, -36, 0, 1, 10, 0, -120, 0, 252, 0, -120, 0, 10, 0, 11, 0, -165, 0, 462, 0, -330, 0, 55, 0, -1, 12, 0, -220, 0, 792, 0, -792, 0, 220, 0, -12, 0, 13, 0, -286, 0, 1287, 0
Offset: 1

Views

Author

Robert G. Wilson v, Jul 06 2004

Keywords

Examples

			The trigonometric expansion of sin(4x) is 4*cos(x)^3*sin(x) - 4*cos(x)*sin(x)^3, so the fourth row is 4, 0, -4, 0.
Triangle begins:
1
2 0
3 0 -1
4 0 -4 0
5 0 -10 0 1
6 0 -20 0 6 0
7 0 -35 0 21 0 -1
8 0 -56 0 56 0 -8 0

Links

Crossrefs

First column is A000027 = C(n, 1), third column is A000292 = C(n, 3), fifth column is A000389 = C(n, 5), seventh column is A000580 = C(n, 7), ninth column is A000582 = C(n, 9).
A001288 = C(n, 11), A010966 = C(n, 13), A010968 = C(n, 15), A010970 = C(n, 17), A010972 = C(n, 19),
A010974 = C(n, 21), A010976 = C(n, 23), A010978 = C(n, 25), A010980 = C(n, 27), A010982 = C(n, 29),
A010984 = C(n, 31), A010986 = C(n, 33), A010988 = C(n, 35), A010990 = C(n, 37), A010992 = C(n, 39),
A010994 = C(n, 41), A010996 = C(n, 43), A010998 = C(n, 45), A011000 = C(n, 47), A017713 = C(n, 49)
Another version of the triangle in A034867. Cf. A096754.
A017715 = C(n, 51), A017717 = C(n, 53), A017719 = C(n, 55), A017721 = C(n, 57), etc.

Programs

  • Mathematica
    Flatten[ Table[ Plus @@ CoefficientList[ TrigExpand[ Sin[n*x]], {Sin[x], Cos[x]}], {n, 13}]]

Formula

T(n,k) = C(n+1,k+1)*sin(Pi*(k+1)/2). - Paul Barry, May 21 2006
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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