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.

Previous Showing 21-30 of 30 results.

A027965 T(n, 2*n-3), T given by A027960.

Original entry on oeis.org

3, 7, 15, 28, 47, 73, 107, 150, 203, 267, 343, 432, 535, 653, 787, 938, 1107, 1295, 1503, 1732, 1983, 2257, 2555, 2878, 3227, 3603, 4007, 4440, 4903, 5397, 5923, 6482, 7075, 7703, 8367, 9068, 9807, 10585, 11403, 12262, 13163, 14107, 15095, 16128, 17207, 18333, 19507, 20730, 22003
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

A column of triangle A027011.

Programs

  • GAP
    List([2..50], n-> (18-10*n+3*n^2+n^3)/6) # G. C. Greubel, Jun 30 2019
  • Magma
    [(18-10*n+3*n^2+n^3)/6: n in [2..50]]; // G. C. Greubel, Jun 30 2019
  • Mathematica
    LinearRecurrence[{4,-6,4,-1}, {3,7,15,28}, 50] (* G. C. Greubel, Jun 30 2019 *)
  • PARI
    vector(50, n, n++; (18-10*n+3*n^2+n^3)/6) \\ G. C. Greubel, Jun 30 2019
  • Sage
    [(18-10*n+3*n^2+n^3)/6 for n in (2..50)] # G. C. Greubel, Jun 30 2019

Formula

a(n+2) = A074742(n-1) = A008778(n) + 2 = A000297(n-1) + 3.
From Ralf Stephan, Feb 07 2004: (Start)
G.f.: x^2*(3 - 2*x)*(1 - x + x^2)/(1-x)^4.
Differences of A027966. (End)
From G. C. Greubel, Jun 30 2019: (Start)
a(n) = (18 - 10*n + 3*n^2 + n^3)/6.
E.g.f.: (-18 - 12*x + (18 - 6*x + 6*x^2 + x^3)*exp(x))/6. (End)

Extensions

Terms a(32) onward added by G. C. Greubel, Jun 30 2019

A096806 Triangle, read by rows, such that the binomial transform of the n-th row lists the m-dimensional partitions of n, for n>=1 and m>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 7, 1, 1, 10, 27, 28, 11, 1, 1, 14, 57, 93, 64, 16, 1, 1, 21, 117, 269, 282, 131, 22, 1, 1, 29, 223, 707, 1062, 766, 244, 29, 1, 1, 41, 417, 1747, 3565, 3681, 1871, 421, 37, 1, 1, 55, 748, 4090, 10999, 15489, 11400, 4152, 683, 46, 1, 1, 76
Offset: 1

Views

Author

Paul D. Hanna, Jul 19 2004

Keywords

Comments

The n-th row equals the inverse binomial transform of n-th column of square array A096751, for n>=1. The zero-dimensional partition of n is taken to be 1 for all n.

Examples

			The number of m-dimensional partitions of 5, for m>=0, is given by the binomial transform of the 5th row:
BINOMIAL([1,6,11,7,1]) = [1,7,24,59,120,216,357,554,819,1165,...] = A008779.
Rows begin:
  [1],
  [1,  1],
  [1,  2,   1],
  [1,  4,   4,    1],
  [1,  6,  11,    7,     1],
  [1, 10,  27,   28,    11,     1],
  [1, 14,  57,   93,    64,    16,      1],
  [1, 21, 117,  269,   282,   131,     22,      1],
  [1, 29, 223,  707,  1062,   766,    244,     29,     1],
  [1, 41, 417, 1747,  3565,  3681,   1871,    421,    37,     1],
  [1, 55, 748, 4090, 10999, 15489,  11400,   4152,   683,    46,    1],
  [1, 76,1326, 9219, 31828, 58975,  59433,  31802,  8483,  1054,   56,   1],
  [1,100,2284,20095, 87490,207735, 276230, 204072, 80664, 16162, 1561,  67, 1],
  [1,134,3898,42707,230737,687665,1173533,1148939,632478,188077,29031,2234,79,1],
  ...
The inverse binomial transform of the diagonals of this triangle begin:
  [1],
  [1, 1,  1],
  [1, 3,  4,   6,  3],
  [1, 5, 16,  29,  49,   45,   15],
  [1, 9, 38, 127, 289,  540,  660,   420, 105],
  [1,13, 90, 397,1384, 3633, 7506, 10920,9765,4725,945],
  [1,20,182,1140,5266,19324,55645,125447,  ? ,  ? , ?  ,62370,10395],
  ...

Links

Crossrefs

Cf. A096751, A096807 (row sums), A000065 (column k=1?), A008778 (bin trans 4th row), A042984 (bin trans 6th row)
Cf. A119271.

Formula

T(n, 0)=T(n, n-1)=1, T(n, 1)=A000041(n)-1, T(n, n-2)=(n-1)*(n-2)/2+1, for n>=1.

A116672 Triangle read by rows in which the binomial transform of the n-th row gives the Euler transform of the n-th diagonal of Pascal's triangle (A007318).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 7, 1, 1, 10, 27, 29, 12, 1, 1, 14, 57, 96, 72, 21, 1, 1, 21, 117, 277, 319, 176, 38, 1
Offset: 1

Views

Author

Alford Arnold, Feb 22 2006

Keywords

Comments

For example, the Euler transform of 1,3,6,... is 1,1,4,10,26,59,141,... (A000294) differing slightly from A000293 which counts the solid partitions.
The NAME does not reproduce the DATA, COMMENTS, or EXAMPLES. - R. J. Mathar, Jul 19 2017
The binomial transforms of the rows form the rows of A289656. - N. J. A. Sloane, Jul 19 2017

Examples

			Row 6 is 1 10 27 29 12 1 generating 1 11 48 141 ... (A008780) the seventh term in the Euler transforms of 1,1,1,...; 1,2,3,...; 1,3,6,... 1,4,10,... etc.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 6, 11, 7, 1;
1, 10, 27, 29, 12, 1;
1, 14, 57, 96, 72, 21, 1;
1, 21, 117, 277, 319, 176, 38, 1;
...

Links

Crossrefs

Cf. A000293, A116673 (row sums), A008778 - A008780, A289656.

A022816 Number of terms in 6th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 11, 44, 121, 271, 532, 952, 1590, 2517, 3817, 5588, 7943, 11011, 14938, 19888, 26044, 33609, 42807, 53884, 67109, 82775, 101200, 122728, 147730, 176605, 209781, 247716, 290899, 339851, 395126, 457312, 527032, 604945, 691747
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

Links

Crossrefs

Cf. A008778, A022811-A022818, A024207-A024210.

Programs

  • Magma
    [n*(n+1)*(n^3+24*n^2+81*n-46)/120: n in [1..40]]; // Vincenzo Librandi, Oct 10 2011
  • Mathematica
    Table[n(n+1)(n^3+24n^2+81n-46)/120,{n,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,44,121,271,532},40] (* Harvey P. Dale, Dec 29 2017 *)
  • PARI
    a(n)=n*(n+1)*(n^3+24*n^2+81*n-46)/120 \\ Charles R Greathouse IV, Oct 21 2022

Formula

a(n) = n*(n+1)*(n^3+24*n^2+81*n-46)/120. G.f.: x*(1+5*x-7*x^2+2*x^3)/(x-1)^6. - R. J. Mathar, Sep 15 2009

Extensions

More terms from Christian G. Bower, Aug 15 1999.

A081498 Consider the triangle in which the n-th row starts with n, contains n terms and the difference of successive terms is 1,2,3,... up to n-1. Sequence gives row sums.

Original entry on oeis.org

1, 3, 5, 6, 5, 1, -7, -20, -39, -65, -99, -142, -195, -259, -335, -424, -527, -645, -779, -930, -1099, -1287, -1495, -1724, -1975, -2249, -2547, -2870, -3219, -3595, -3999, -4432, -4895, -5389, -5915, -6474, -7067, -7695, -8359, -9060, -9799, -10577, -11395, -12254, -13155, -14099, -15087, -16120
Offset: 1

Views

Author

Amarnath Murthy, Mar 25 2003

Keywords

Comments

The triangle whose row sums are being considered is:
1;
2, 1;
3, 2, 0;
4, 3, 1, -2;
5, 4, 2, -1, -5;
6, 5, 3, 0, -4, -9;
7, 6, 4, 1, -3, -8, -14;
The leading diagonal is given by A080956(n-1) = n*(3-n)/2.

Examples

			G.f. = x * (1 + 3*x + 5*x^2 + 6*x^3 + 5*x^4 + x^5 - 7*x^6 - 20*x^7 - 39*x^8 - 65*x^9 + ...).

Links

Crossrefs

Cf. A008778, A080956, A081499.

Programs

  • GAP
    List([1..50],n->n^2-Binomial(n+1,n-2)); # Muniru A Asiru, Mar 05 2019
  • Magma
    [n*(1+6*n-n^2)/6: n in [1..50]]; // G. C. Greubel, Mar 06 2019
  • Maple
    seq(n^2-binomial(n+1,n-2),n=1..50); # C. Ronaldo
[seq(binomial(n,2)+binomial(n,1)-binomial(n,3), n=1..49)]; # Zerinvary Lajos, Jul 23 2006
  • Mathematica
    LinearRecurrence[{4,-6,4,-1}, {1,3,5,6}, 50] (* G. C. Greubel, Mar 06 2019 *)
  • PARI
    {a(n) = if( n< 0, n = -2 - n; polcoeff( (1 + x - x^2) / (1 - x)^4 + x * O(x^n), n), polcoeff( (1 - x - x^2) / (1 - x)^4 + x * O(x^n), n))} /* Michael Somos, Jul 04 2012 */
  • PARI
    vector(50, n, n*(1+6*n-n^2)/6) \\ G. C. Greubel, Mar 06 2019
  • Sage
    [n*(1+6*n-n^2)/6 for n in (1..50)] # G. C. Greubel, Mar 06 2019

Formula

a(n) = n^2 - binomial(n+1, n-2). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
a(n) = binomial(n,2)+binomial(n,1)-binomial(n,3). - Zerinvary Lajos, Jul 23 2006
a(n) = n*(1+6*n-n^2)/6. - Karen A. Yeats, Nov 20 2006
From Michael Somos, Jul 04 2012: (Start)
G.f.: x * (1 - x - x^2) / (1 - x)^4.
a(-1 - n) = A008778(n). (End)
E.g.f.: x*(6 +3*x -x^2)*exp(x)/6. - G. C. Greubel, Mar 06 2019

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
Offset changed to 1 at the suggestion of Michel Marcus, Mar 05 2019
Formulas and programs addapted for offset 1 by Michel Marcus, Mar 05 2019

A108446 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k peaks of the form ud.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 20, 32, 13, 1, 113, 223, 135, 26, 1, 688, 1620, 1300, 412, 45, 1, 4404, 12064, 12050, 5350, 1030, 71, 1, 29219, 91335, 109134, 62450, 17575, 2247, 105, 1, 199140, 699689, 973077, 682234, 254625, 49210, 4438, 148, 1, 1385904, 5407744
Offset: 0

Views

Author

Emeric Deutsch, Jun 10 2005

Keywords

Comments

Row sums yield A027307. Column 0 yields A108447. T(n,n-1) = A008778(n-1) = n(n^2+6n-1)/6. Number of ud peaks in all paths from (0,0) to (3n,0) is given by A108448.

Examples

			T(2,1) = 5 because we have udUdd, uudd, Uddud, Ududd and Uuddd.
Triangle begins:
1;
1,1;
4,5,1;
20,32,13,1;
113,223,135,26,1;

Links

Crossrefs

Cf. A027307, A008778, A108447, A108448, A108425, A108426.

Programs

  • Maple
    T:=proc(n,k) if n=0 and k=0 then 1 elif n=0 then 0 elif k=n then 1 elif k=n then 1 else (1/n)*binomial(n,k)*sum(binomial(n-k,j)*binomial(n+2*j,k+j-1),j=0..n-k) fi end: for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
  • Mathematica
    T[0, 0] = 1; T[n_, k_] := (1/n) Binomial[n, k]*Sum[Binomial[n-k, j]* Binomial[n+2j, k+j-1], {j, 0, n-k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018 *)

Formula

T(n,k) = (1/n) binomial(n, k)*sum(binomial(n-k,j)*binomial(n+2j,k+j-1), j=0..n-k).
G.f.: G = G(t,z) satisfies G = 1+z(G-1+t)G+zG^3.

A289656 Triangle read by rows: row n gives the first n terms of the binomial transform of the n-th row of A116672.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 5, 13, 26, 1, 7, 24, 59, 120, 1, 11, 48, 141, 331, 672, 1, 15, 86, 310, 855, 1982, 4067, 1, 22, 160, 692, 2214, 5817, 13301, 27428
Offset: 1

Views

Author

N. J. A. Sloane, Jul 19 2017

Keywords

Comments

Rows 4, 5, 6 match the starts of sequences A008778, A008779, A008780.

Examples

			Triangle begins:
[1]
[1, 2]
[1, 3, 6]
[1, 5, 13, 26]
[1, 7, 24, 59, 120]
[1, 11, 48, 141, 331, 672]
[1, 15, 86, 310, 855, 1982, 4067]
[1, 22, 160, 692, 2214, 5817, 13301, 27428]
...

Links

Crossrefs

Cf. A116672, A008778, A008779, A008780.

A271702 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S2(k,j), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 10, 26, 71, 1, 5, 15, 45, 140, 456, 1, 6, 21, 71, 246, 887, 3337, 1, 7, 28, 105, 399, 1568, 6405, 27203, 1, 8, 36, 148, 610, 2584, 11334, 51564, 243203, 1, 9, 45, 201, 891, 4035, 18849, 91101, 455712, 2357356
Offset: 0

Views

Author

Peter Luschny, Apr 14 2016

Keywords

Examples

			Triangle starts:
[1]
[1, 1]
[1, 2, 3]
[1, 3, 6, 13]
[1, 4, 10, 26, 71]
[1, 5, 15, 45, 140, 456]
[1, 6, 21, 71, 246, 887, 3337]
[1, 7, 28, 105, 399, 1568, 6405, 27203]

Crossrefs

A000012 (col. 0), A000027 (col. 1), A000217 (col. 2), A008778 (col. 3), A122455 (diag. n,n), A134094 (diag. n,n-1).
Cf. A048993.

Programs

  • Maple
    T := (n,k) -> add(Stirling2(k,j)*binomial(-j-1,-n-1)*(-1)^(n-j),j=0..n):
seq(seq(T(n,k), k=0..n), n=0..9);
  • Mathematica
    Flatten[Table[Sum[(-1)^(n-j) Binomial[-j-1,-n-1] StirlingS2[k,j], {j,0,n}], {n,0,9}, {k,0,n}]]

Formula

T(n,k) = Sum_{j=0..k} C(n,j) * S2(k,j). - Alois P. Heinz, Sep 03 2019

A127738 Triangle read by rows: the matrix product A004736 * A127701 of two triangular matrices.

Original entry on oeis.org

1, 3, 2, 5, 5, 3, 7, 8, 7, 4, 9, 11, 11, 9, 5, 11, 14, 15, 14, 11, 6, 13, 17, 19, 19, 17, 13, 7, 15, 20, 23, 24, 23, 20, 15, 8, 17, 23, 27, 29, 29, 27, 23, 17, 9, 19, 26, 31, 34, 35, 34, 31, 26, 19, 10
Offset: 1

Views

Author

Gary W. Adamson, Jan 27 2007

Keywords

Comments

Left column = A028387: (1, 5, 11, 19, 29, 41, 55, ...).

Examples

			First few rows of the triangle:
   1;
   3,  2;
   5,  5,  3;
   7,  8,  7,  4;
   9, 11, 11,  9,  5;
  11, 14, 15, 14, 11,  6;
  13, 17, 19, 19, 17, 13,  7;
  ...

Crossrefs

Cf. A004736, A127701, A008778 (row sums), A028387.

Formula

T(n,k) = Sum_{j=k..n} A004736(n,j)*A127701(j,k). - R. J. Mathar, Aug 31 2022
T(n,k) = k+(k+1)*(n-k) = n+k*(n-k) = n +A094053(n,k) = A059036(n,k). - R. J. Mathar, Aug 31 2022

A377069 Triangle read by rows: T(n,k) is the number of (k+1)-vertex dominating sets of the (n+1)-path graph that include the first vertex.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 3, 1, 0, 1, 5, 4, 1, 0, 0, 5, 9, 5, 1, 0, 0, 3, 13, 14, 6, 1, 0, 0, 1, 13, 26, 20, 7, 1, 0, 0, 0, 9, 35, 45, 27, 8, 1, 0, 0, 0, 4, 35, 75, 71, 35, 9, 1, 0, 0, 0, 1, 26, 96, 140, 105, 44, 10, 1, 0, 0, 0, 0, 14, 96, 216, 238, 148, 54, 11, 1
Offset: 0

Views

Author

Andrew Howroyd, Oct 21 2024

Keywords

Comments

T(n,k) is also the number of (k+1)-vertex dominating sets of the (n+2)-path graph that do not include the first vertex.

Examples

			Triangle begins:
  1;
  1, 1;
  0, 2, 1;
  0, 2, 3,  1;
  0, 1, 5,  4,  1;
  0, 0, 5,  9,  5,  1;
  0, 0, 3, 13, 14,  6,   1;
  0, 0, 1, 13, 26, 20,   7,   1;
  0, 0, 0,  9, 35, 45,  27,   8,  1;
  0, 0, 0,  4, 35, 75,  71,  35,  9,  1;
  0, 0, 0,  1, 26, 96, 140, 105, 44, 10, 1;
  ...
Corresponding to T(4,2) = 5, a path graph with 5 vertices has the following 3-vertex dominating sets that include the first vertex (x marks a vertex in the set):
   x . . x x
   x . x . x
   x . x x .
   x x . . x
   x x . x .

Links

Crossrefs

Row sums are A047081.
Column sums are A008776.
Diagonals include A000012, A000027, A000096, A008778, A095661.
Cf. A169623, A212633.

Programs

  • PARI
    T(n)={[Vecrev(p) | p<-Vec((1 + x)/(1 - y*x - y*x^2 - y*x^3) + O(x*x^n))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

Formula

G.f.: (1 + x)/(1 - y*x - y*x^2 - y*x^3).
A212633(n,k) = T(n-1, k-1) + T(n-2, k-1).
Previous Showing 21-30 of 30 results.

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

Content is available under The OEIS End-User License Agreement.