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.

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 5, 4, 1, 4, 12, 13, 6, 1, 8, 28, 38, 25, 8, 1, 16, 64, 104, 88, 41, 10, 1, 32, 144, 272, 280, 170, 61, 12, 1, 64, 320, 688, 832, 620, 292, 85, 14, 1, 128, 704, 1696, 2352, 2072, 1204, 462, 113, 16, 1
Offset: 0

Views

Author

Philippe Deléham, Dec 05 2011

Keywords

Comments

Diagonals ascending: 1, 0, 1, 1, 2, 2, 4, 5, 1, 8, 12, 4, ... (see A201509).

Examples

			Triangle begins:
  1;
  0,  1;
  1,  2,  1;
  2,  5,  4,  1;
  4, 12, 13,  6,  1;
  8, 28, 38, 25,  8,  1;

Links

Crossrefs

Diagonals: A000012, A005843, A001844, A035597,
Columns: A034008, A045623, A084851, A055585,
Row sums: A052156

Programs

  • Mathematica
    CoefficientList[#, y]& /@ CoefficientList[(1-x)^2/(1-(y+2)*x) + O[x]^10, x] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

Formula

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 0, T(1,0) = 0, T(2,0) = 0 and T(n,k)= 0 if k < 0 or if n < k.
Sum_{k=0..n} T(n,k)*x^k = A154955(n+1), A034008(n), A052156(n), A055841(n), A055842(n), A055846(n), A055270(n), A055847(n), A055995(n), A055996(n), A056002(n), A056116(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10 respectively.
G.f.: (1-x)^2/(1-(y+2)*x).

A055584 Triangle of partial row sums (prs) of triangle A055252.

Original entry on oeis.org

1, 5, 1, 19, 6, 1, 63, 25, 7, 1, 192, 88, 32, 8, 1, 552, 280, 120, 40, 9, 1, 1520, 832, 400, 160, 49, 10, 1, 4048, 2352, 1232, 560, 209, 59, 11, 1, 10496, 6400, 3584, 1792, 769, 268, 70, 12, 1, 26624, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 66304, 43520
Offset: 0

Views

Author

Wolfdieter Lang, May 26 2000

Keywords

Comments

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is (((1-z)^3)/(1-2*z)^4)/(1-x*z/(1-z)).
This is the fourth member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
The column sequences appear as A049612(n+1), A055585, A001794, A001789(n+3), A027608, A055586 for m=0..5.

Examples

			[0] 1
[1] 5, 1
[2] 19, 6, 1
[3] 63, 25, 7, 1
[4] 192, 88, 32, 8, 1
[5] 552, 280, 120, 40, 9, 1
[6] 1520, 832, 400, 160, 49, 10, 1
[7] 4048, 2352, 1232, 560, 209, 59, 11, 1
Fourth row polynomial (n=3): p(3, x)= 63 + 25*x + 7*x^2 + x^3.

Crossrefs

Cf. A007318, A055248, A055249, A055252. Row sums: A049600(n+1, 4).

Programs

  • Maple
    T := (n, k) -> binomial(n, k)*hypergeom([4, k - n], [k + 1], -1):
for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # Peter Luschny, Sep 23 2024

Formula

a(n, m)=sum(A055252(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n
Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A055252(n, m), n >= m >= 0, a(n, m) := 0 if n
G.f. for column m: (((1-x)^3)/(1-2*x)^4)*(x/(1-x))^m, m >= 0.
T(n, k) = binomial(n, k)*hypergeom([4, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

A100313 Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (10;0) and (01;1).

Original entry on oeis.org

1, 16, 96, 400, 1408, 4480, 13312, 37632, 102400, 270336, 696320, 1757184, 4358144, 10649600, 25690112, 61276160, 144703488, 338690048, 786432000, 1812987904, 4152360960, 9453961216, 21407727616, 48234496000, 108179488768, 241591910400, 537407782912
Offset: 0

Views

Author

Sergey Kitaev, Nov 13 2004

Keywords

Comments

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by the g.f. 2*x*y/(1-2*(x+y-x*y)).

Links

Crossrefs

Cf. A055585, A100312 (m=3), this sequence (m=4).

Programs

  • Magma
    [2^n*n*(n^2+9*n+14)/3 +0^n: n in [0..40]]; // G. C. Greubel, Feb 01 2023
  • Mathematica
    Table[If[n==0, 1, 2^n*n*(n^2+9*n+14)/3], {n,0,40}] (* G. C. Greubel, Feb 01 2023 *)
  • PARI
    vector(50, n, n*(n^2+9*n+14) * 2^n / 3) \\ Michel Marcus, Dec 01 2014
  • SageMath
    [2^n*n*(n^2+9*n+14)/3 +0^n for n in range(41)] # G. C. Greubel, Feb 01 2023

Formula

G.f.: 1 + 16*x*(1-x)^2/(1-2*x)^4.
a(n) = (1/3) n*(n^2 + 9*n + 14) * 2^n for n>0, with a(0) = 1.
a(n) = 16 * A055585(n-1) for n>0.
E.g.f.: (1/3)*(3 + 8*x*(6 + 6*x + x^2)*exp(2*x)). - G. C. Greubel, Feb 01 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Dec 21 2018

A375550 Triangle read by rows: T(m, n, k) = binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1) for m = 4.

Original entry on oeis.org

1, 6, 1, 25, 7, 1, 88, 32, 8, 1, 280, 120, 40, 9, 1, 832, 400, 160, 49, 10, 1, 2352, 1232, 560, 209, 59, 11, 1, 6400, 3584, 1792, 769, 268, 70, 12, 1, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1
Offset: 0

Views

Author

Peter Luschny, Sep 23 2024

Keywords

Comments

Triangle T(m,n,k) is a Riordan array of the form ((1-x)^(m-1)*(1-2x)^(-m-1), x/(1-x)), for m = 3. - Igor Victorovich Statsenko, Feb 08 2025

Examples

			Triangle starts:
  [0]     1;
  [1]     6,     1;
  [2]    25,     7,     1;
  [3]    88,    32,     8,    1;
  [4]   280,   120,    40,    9,    1;
  [5]   832,   400,   160,   49,   10,    1;
  [6]  2352,  1232,   560,  209,   59,   11,   1;
  [7]  6400,  3584,  1792,  769,  268,   70,  12,  1;
  [8] 16896,  9984,  5376, 2561, 1037,  338,  82, 13,  1;
  [9] 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1;
  ...
Seen as an array of the columns:
  [0] 1,  6, 25,  88,  280,  832,  2352,  6400,  16896, ...
  [1] 1,  7, 32, 120,  400, 1232,  3584,  9984,  26880, ...
  [2] 1,  8, 40, 160,  560, 1792,  5376, 15360,  42240, ...
  [3] 1,  9, 49, 209,  769, 2561,  7937, 23297,  65537, ...
  [4] 1, 10, 59, 268, 1037, 3598, 11535, 34832, 100369, ...
  [5] 1, 11, 70, 338, 1375, 4973, 16508, 51340, 151709, ...
  [6] 1, 12, 82, 420, 1795, 6768, 23276, 74616, 226325, ...

Crossrefs

Column k: A055585 (k=0), A001794 (k=1), A001789 (k=2), A027608 (k=3), A055586 (k=4).
Cf. A145018 (diagonal n-2), A375549 (row sums), A049612 (alternating row sums), A122433.

Programs

  • Maple
    T := (m, n, k) -> binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1);
for n from 0 to 9 do seq(simplify(T(4, n, k)), k = 0..n) od;
# As a binomial sum:
T := (m, n, k) -> add(binomial(m + j, m)*binomial(n + 1, n - (j + k)), j = 0..n-k):
for n from 0 to 9 do [n], seq(T(3, n, k), k = 0..n) od;
# Alternative, generating the array of the columns:
cgf := k -> (1 - x)^(2 - k) / (1 - 2*x)^4:
ser := (k, len) -> series(cgf(k), x, len + 2):
Tcol := (k, len) -> seq(coeff(ser(k, len), x, j), j = 0..len):
seq(lprint([k], Tcol(k, 8)), k = 0..6);

Formula

T(m, n, k) = Sum_{j=0..n-k} binomial(m + j, m)*binomial(n + 1, n - (j + k)) for m = 3.
G.f. of column k: (1 - x)^(2 - k) / (1 - 2*x)^4.

A125091 Triangle read by rows: T(n,k) = (1/6)*k*(k+1)*(k+2)*binomial(n,k) (1 <= k <= n).

Original entry on oeis.org

1, 2, 4, 3, 12, 10, 4, 24, 40, 20, 5, 40, 100, 100, 35, 6, 60, 200, 300, 210, 56, 7, 84, 350, 700, 735, 392, 84, 8, 112, 560, 1400, 1960, 1568, 672, 120, 9, 144, 840, 2520, 4410, 4704, 3024, 1080, 165, 10, 180, 1200, 4200, 8820, 11760, 10080, 5400, 1650, 220, 11
Offset: 1

Views

Author

Gary W. Adamson, Nov 19 2006

Keywords

Comments

T(n,n) = n*(n+1)*(n+2)/6 = A000292(n).
Sum_{k=1..n} T(n,k) = 2^n*n*(n+2)*(n+7)/48 = A055585(n-1).

Examples

			Triangle starts:
  1;
  2,   4;
  3,  12,  10;
  4,  24,  40,  20;
  5,  40, 100, 100,  35;
  6,  60, 200, 300, 210,  56;
  7,  84, 350, 700, 735, 392,  84;

Crossrefs

Cf. A055585.
Cf. A000292.

Programs

  • Maple
    T:=(n,k)->k*(k+1)*(k+2)*binomial(n,k)/6: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
  • Mathematica
    Flatten[Table[(k(k+1)(k+2)Binomial[n,k])/6,{n,20},{k,n}]] (* Harvey P. Dale, Jan 23 2016 *)

Extensions

Edited by N. J. A. Sloane, Dec 04 2006

A272099 Triangle read by rows, T(n,k) = C(n+1,k+1)*F([k-n, k-n-1], [-n-1], -1), where F is the generalized hypergeometric function, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 4, 1, 12, 5, 1, 32, 18, 6, 1, 80, 56, 25, 7, 1, 192, 160, 88, 33, 8, 1, 448, 432, 280, 129, 42, 9, 1, 1024, 1120, 832, 450, 180, 52, 10, 1, 2304, 2816, 2352, 1452, 681, 242, 63, 11, 1, 5120, 6912, 6400, 4424, 2364, 985, 316, 75, 12, 1
Offset: 0

Views

Author

Peter Luschny, Apr 25 2016

Keywords

Comments

This triangle results when the first column is removed from A210038. - Georg Fischer, Jul 26 2023

Examples

			Triangle starts:
1;
4,    1;
12,   5,    1;
32,   18,   6,   1;
80,   56,   25,  7,   1;
192,  160,  88,  33,  8,   1;
448,  432,  280, 129, 42,  9,  1;
1024, 1120, 832, 450, 180, 52, 10, 1;

Crossrefs

A258109 (row sums), A008466 (alternating row sums), A001787 (col. 0), A001793 (col. 1), A055585 (col. 2).
Cf. A210038.

Programs

  • Maple
    T := (n,k) -> binomial(n+1,k+1)*hypergeom([k-n, k-n-1], [-n-1], -1):
seq(seq(simplify(T(n,k)),k=0..n),n=0..9);
  • Mathematica
    T[n_, k_] := Binomial[n+1, k+1] HypergeometricPFQ[{k-n, k-n-1}, {-n-1}, -1];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 22 2019 *)
Showing 1-6 of 6 results.

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

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