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 1-6 of 6 results.

A060102 Bisection of triangle A060098: even-indexed members of column sequences of A060098 (not counting leading zeros).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 8, 1, 1, 16, 30, 13, 1, 1, 25, 80, 71, 19, 1, 1, 36, 175, 259, 140, 26, 1, 1, 49, 336, 742, 660, 246, 34, 1, 1, 64, 588, 1806, 2370, 1442, 399, 43, 1, 1, 81, 960, 3906, 7062, 6292, 2828, 610, 53
Offset: 0

Views

Author

Wolfdieter Lang, Apr 06 2001

Keywords

Comments

Row sums give A052975. Column sequences without leading zeros give for m=0..5: A000012 (powers of 1), A000290 (squares), A002417(n+1), A060103-5.
Companion triangle (odd-indexed members) A060556.

Examples

			{1}; {1,1}; {1,4,1}; {1,9,8,1}; ... Pe(3,x) = 1 + 3*x.

Formula

a(n, m) = A060098(2*n-m, m).
a(n, m) = Sum_{j=0..floor((m+1)/2)} binomial((n-m)-j+2*m, 2*m)*binomial(m+1, 2*j), n >= m >= 0, otherwise zero.
G.f. for column m: (x^m)*Pe(m+1, x)/(1-x)^(2*m+1), with Pe(n, x) = Sum_{j=0..floor(n/2)} binomial(n, 2*j)*x^j (even members of row n of Pascal triangle A007318).

A060556 Bisection of triangle A060098: odd-indexed members of column sequences of A060098 (not counting leading zeros).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 12, 16, 4, 1, 20, 50, 32, 5, 1, 30, 120, 140, 55, 6, 1, 42, 245, 448, 316, 86, 7, 1, 56, 448, 1176, 1284, 622, 126, 8, 1, 72, 756, 2688, 4170, 3102, 1113, 176, 9, 1, 90, 1200, 5544, 11550, 12122
Offset: 0

Views

Author

Wolfdieter Lang, Apr 06 2001

Keywords

Comments

Row sums give A060557. Column sequences without leading zeros give for m=0..5: A000012 (powers of 1), A002378 = 2*A000217, A004320, 4*A040977, A060558, 2*A060559.
Companion triangle (even-indexed members) A060102.
With offset 1 for n and k, T(n,k) is the number of (1-2-3)-avoiding trapezoidal words of length n that contain n+1-k 1s. A trapezoidal word (following Riordan) is a sequence (a_1,a_2,...,a_n) of integers with 1 <= a_i <= 2i-1. For example, T(3,3)=3 counts 122, 132, 133 and T(4,2)=12 counts 1112, 1113, 1114, 1115, 1116, 1117, 1121, 1131, 1141, 1151, 1211, 1311. - David Callan, Aug 25 2009

Examples

			{1}; {1,2}; {1,6,3}; {1,12,16,4}; ...; Po(3,x) = 3 + x.

Formula

a(n, m)= A060098(2*n+1-m, m).
G.f. for column m: (x^m)*Po(m+1, x)/(1-x)^(2*m+1), with Po(n, x) = Sum_{j=0..floor(n/2)} binomial(n, 2*j+1)*x^j (odd members of row n of Pascal triangle A007318).
a(n, m) = Sum_{j=0..floor((m+1)/2)} binomial(n-j+m, 2*m)*binomial(m+1, 2*j+1), n >= m >= 0, otherwise zero.

A060100 Fifth column (m=4) of triangle A060098.

Original entry on oeis.org

1, 5, 19, 55, 140, 316, 660, 1284, 2370, 4170, 7062, 11550, 18348, 28380, 42900, 63492, 92235, 131703, 185185, 256685, 351208, 474760, 634712, 839800, 1100580, 1429428, 1841100, 2352732, 2984520, 3759720, 4705464, 5852760, 7237461, 8900265, 10887855
Offset: 0

Views

Author

Wolfdieter Lang, Apr 06 2001

Keywords

Comments

Partial sums of A038164.

Links

Programs

  • PARI
    Vec(1 / ((1-x)^9*(1+x)^4) + O(x^40)) \\ Colin Barker, Jan 17 2017

Formula

a(n)= sum(A060098(n+4, 4)).
G.f.: 1/((1-x^2)^4*(1-x)^5) = 1/((1-x)^9*(1+x)^4).
a(n) = (315*(3797+299*(-1)^n) + 12*(204347+4165*(-1)^n)*n + 2*(970241+4095*(-1)^n)*n^2 + 28*(28457+15*(-1)^n)*n^3 + 189168*n^4 + 26936*n^5 + 2268*n^6 + 104*n^7 + 2*n^8) / 1290240. - Colin Barker, Jan 17 2017

A060101 Sixth column (m=5) of triangle A060098.

Original entry on oeis.org

1, 6, 26, 86, 246, 622, 1442, 3102, 6292, 12122, 22374, 39754, 68354, 114114, 185614, 294866, 458601, 699556, 1048476, 1546116, 2246244, 3218644, 4553484, 6365684, 8801104, 12042732, 16319252, 21913612, 29174652, 38528732, 50495236, 65702076, 84906041
Offset: 0

Views

Author

Wolfdieter Lang, Apr 06 2001

Keywords

Comments

Partial sums of A038165.

Links

Programs

  • Mathematica
    Accumulate[CoefficientList[Series[1/((1-x)(1-x^2))^5,{x,0,35}],x]] (* or *) LinearRecurrence[ {6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1},{1,6,26,86,246,622,1442,3102,6292,12122,22374,39754,68354,114114,185614,294866},30] (* Harvey P. Dale, Mar 06 2016 *)
  • PARI
    Vec(1/ ((1-x)^11*(1+x)^5) + O(x^40)) \\ Colin Barker, Jan 17 2017

Formula

a(n)= sum(A060098(n+5, 5)).
G.f.: 1/((1-x^2)^5*(1-x)^6) = 1/((1-x)^11*(1+x)^5).
a(n) = (14175*(30827+1941*(-1)^n) + 1440*(676427+11445*(-1)^n)*n + 126*(6861329+27375*(-1)^n)*n^2 + 1600*(258451+189*(-1)^n)*n^3 + 10*(12016607+945*(-1)^n)*n^4 + 22444800*n^5 + 2754192*n^6 + 220800*n^7 + 11130*n^8 + 320*n^9 + 4*n^10)/ 464486400. - Colin Barker, Jan 17 2017

A060099 G.f.: 1/((1-x^2)^3*(1-x)^4).

Original entry on oeis.org

1, 4, 13, 32, 71, 140, 259, 448, 742, 1176, 1806, 2688, 3906, 5544, 7722, 10560, 14223, 18876, 24739, 32032, 41041, 52052, 65429, 81536, 100828, 123760, 150892, 182784, 220116, 263568, 313956, 372096, 438957, 515508, 602889, 702240, 814891, 942172, 1085623
Offset: 0

Views

Author

Wolfdieter Lang, Apr 06 2001

Keywords

Comments

Fourth column (m=3) of triangle A060098.
Partial sums of A038163.
Equals the tetrahedral numbers, [1, 4, 10, 20, ...] convolved with the aerated triangular numbers, [1, 0, 3, 0, 6, 0, 10, ...]. [Gary W. Adamson, Jun 11 2009]

References

  • B. Broer, Hilbert series for modules of covariants, in Algebraic Groups and Their Generalizations..., Proc. Sympos. Pure Math., 56 (1994), Part I, 321-331. See p. 329.

Links

Crossrefs

Cf. A002620, A002624, A096338.
Cf. A001752 (for the similar series 1/((1-x)^4*(1-x^2))).
Cf. A028346 (for the similar series 1/((1-x)^4*(1-x^2)^2)).

Programs

  • Mathematica
    a[n_]:=If[OddQ[n],((1+n) (3+n) (5+n)^2 (7+n) (9+n))/5760,((2+n) (4+n) (6+n) (8+n) (15+10 n+n^2))/5760]; Map[a,Range[0,100]] (* Peter J. C. Moses, Mar 24 2013 *)
CoefficientList[Series[1/((1-x^2)^3*(1-x)^4),{x,0,100}],x] (* Peter J. C. Moses, Mar 24 2013 *)
LinearRecurrence[{4,-3,-8,14,0,-14,8,3,-4,1},{1,4,13,32,71,140,259,448,742,1176},40] (* Harvey P. Dale, Apr 06 2018 *)

Formula

a(n) = Sum_{} A060098(n+3, 3).
G.f.: 1/((1-x)^7*(1+x)^3).

A188314 Expansion of (1/(1-x))*c(x/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 5, 16, 57, 219, 883, 3687, 15803, 69128, 307363, 1385003, 6310869, 29028616, 134610771, 628612921, 2953640371, 13953726888, 66240021987, 315812059436, 1511569447859, 7260364084997, 34984937594741, 169073568381936, 819288294835939, 3979892232651125, 19377475499900015
Offset: 0

Views

Author

Paul Barry, Mar 28 2011

Keywords

Comments

Hankel transform is the (25,-29) Somos-4 sequence A188315. Image of the Catalan numbers by A060098.

Links

Crossrefs

Cf. A000108, A188312.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^2- Sqrt(1-4*x-6*x^2+x^4))/(2*x))); // G. C. Greubel, Aug 14 2018
  • Mathematica
    CoefficientList[Series[(1-x^2 - Sqrt[1-4*x-6*x^2+x^4])/(2*x), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)
  • PARI
    x='x+O('x^30); Vec((1-x^2- sqrt(1-4*x-6*x^2+x^4))/(2*x)) \\ G. C. Greubel, Aug 14 2018

Formula

G.f.: (1-x^2- sqrt(1-4*x-6*x^2+x^4))/(2*x).
G.f.: (1+x)/(1-x^2-x/(1-x-x/(1-x^2-x/(1-x-x/(1-...))))) (continued fraction).
a(n) = Sum{k=0..n, A000108(k)*Sum{i=0..floor(n/2), C(n-2i,n-2i-k)*C(k+i-1,i)}}.
Conjecture: (n+1)*a(n) +(n+2)*a(n-1) +(42-26*n)*a(n-2) +30*(3-n)*a(n-3) +(n-5)*a(n-4) +5*(n-6)*a(n-5)=0. - R. J. Mathar, Nov 15 2011
G.f. A(x) satisfies: A(x) = 1 + x * (1 + x*A(x) + A(x)^2). - Ilya Gutkovskiy, Jul 01 2020
Showing 1-6 of 6 results.

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