A248324 -id:A248324 - cp's OEIS frontend

A248325 Square array read by antidiagonals downwards: super Patalan numbers of order 4.

1, 4, 12, 40, 24, 168, 480, 160, 224, 2464, 6240, 1440, 1120, 2464, 36960, 84864, 14976, 8064, 9856, 29568, 561792, 1188096, 169728, 69888, 59136, 98560, 374528, 8614144, 16972800, 2036736, 678912, 439296, 506880, 1070080, 4922368, 132903936, 246105600, 25459200, 7128576, 3734016, 3294720, 4815360

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers of Gessel, A068555, based on Patalan numbers of order 4, A025749.

			T(0..4, 0..4) is:
  1      4      40     480    6240
  12     24     160    1440   14976
  168    224    1120   8064   69888
  2464   2464   9856   59136  439296
  36960  29568  98560  506880 3294720

Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.

Cf. A068555, A025749, A216702 (first column), A034385 (first row), A248324.

T(0,0)=1, T(n,k) = T(n-1,k)*(16*n-4)/(n+k), T(n,k) = T(n,k-1)*(16*k-12)/(n+k).

G.f.: (x/(1-16*x)^(3/4)+y/(1-16*y)^(1/4))/(x+y-16*x*y).

A248326 Square array read by downward antidiagonals: super Patalan numbers of order 5.

Original entry on oeis.org

1, 5, 20, 75, 50, 450, 1375, 500, 750, 10500, 27500, 6875, 5625, 13125, 249375, 577500, 110000, 61875, 78750, 249375, 5985000, 12512500, 1925000, 825000, 721875, 1246875, 4987500, 144637500, 277062500, 35750000, 12375000, 8250000, 9796875, 21375000, 103312500, 3512625000, 6233906250, 692656250

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers of Gessel, A068555, based on Patalan numbers of order 5, A025750.

			T(0..4,0..4) is
  1       5       75      1375    27500
  20      50      500     6875    110000
  450     750     5625    61875   825000
  10500   13125   78750   721875  8250000
  249375  249375  1246875 9796875 97968750

Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.

Cf. A068555, A025750, A034688 (first row), A049382 (first column), A248324, A248325, A248328, A248329, A248332.

matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*25^(n+k)*binomial(n-1/5,n+k)) \\ Michel Marcus, Oct 09 2014

T(0,0)=1, T(n,k) = T(n-1,k)*(25*n-5)/(n+k), T(n,k) = T(n,k-1)*(25*k-20)/(n+k).
G.f.: (x/(1-25*x)^(4/5)+y/(1-25*y)^(1/5))/(x+y-25*x*y).
T(n,k) = (-1)^k*25^(n+k)*binomial(n-1/5,n+k).

A248328 Square array read by antidiagonals downwards: super Patalan numbers of order 6.

Original entry on oeis.org

1, 6, 30, 126, 90, 990, 3276, 1260, 1980, 33660, 93366, 24570, 20790, 50490, 1161270, 2800980, 560196, 324324, 424116, 1393524, 40412196, 86830380, 14004900, 6162156, 5513508, 9754668, 40412196, 1414426860, 2753763480, 372130200, 132046200, 89791416, 108694872, 242473176, 1212365880

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers, A068555, based on Patalan numbers of order 6, A025751.

			T(0..4,0..4) is
  1          6         126       3276      93366
  30         90        1260      24570     560196
  990        1980      20790     324324    6162156
  33660      50490     424116    5513508   89791416
  1161270    1393524   9754668   108694872 1548901926

Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.

Cf. A068555, A025751, A004993 (first row), A004994 (first column), A004995 (second row), A004996 (second column), A248324, A248325, A248326, A248329, A248332.

matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*36^(n+k)*binomial(n-1/6,n+k)) \\ Michel Marcus, Oct 09 2014

T(0,0)=1, T(n,k) = T(n-1,k)*(36*n-6)/(n+k), T(n,k) = T(n,k-1)*(36*k-30)/(n+k).
G.f.: (x/(1-36*x)^(5/6)+y/(1-36*y)^(1/6))/(x+y-36*x*y).
T(n,k) = (-1)^k*36^(n+k)*binomial(n-1/6,n+k).

A248329 Square array read by antidiagonals downwards: super Patalan numbers of order 7.

Original entry on oeis.org

1, 7, 42, 196, 147, 1911, 6860, 2744, 4459, 89180, 264110, 72030, 62426, 156065, 4213755, 10722866, 2218524, 1310946, 1747928, 5899257, 200574738, 450360372, 75060062, 33647614, 30588740, 55059732, 234003861, 9594158301, 19365495996, 2702162232, 975780806, 672952280, 825895980, 1872030888

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers, A068555, based on Patalan numbers of order 7, A025752.

			T(0..4,0..4) is
  1           7          196        6860       264110
  42          147        2744       72030      2218524
  1911        4459       62426      1310946    33647614
  89180       156065     1747928    30588740   672952280
  4213755     5899257    55059732   825895980  15898497615

Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.

Cf. A068555, A025752, A034835 (first row), A216703 (first column), A248324, A248325, A248326, A248328, A248332.

matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*49^(n+k)*binomial(n-1/7,n+k)) \\ Michel Marcus, Oct 09 2014

T(0,0)=1, T(n,k) = T(n-1,k)*(49*n-7)/(n+k), T(n,k) = T(n,k-1)*(49*k-42)/(n+k).
G.f.: (x/(1-49*x)^(6/7)+y/(1-49*y)^(1/7))/(x+y-49*x*y).
T(n,k) = (-1)^k*49^(n+k)*binomial(n-1/7,n+k).

A248332 Square array read by antidiagonals downwards: super Patalan numbers of order 8.

Original entry on oeis.org

1, 8, 56, 288, 224, 3360, 13056, 5376, 8960, 206080, 652800, 182784, 161280, 412160, 12776960, 34467840, 7311360, 4386816, 5935104, 20443136, 797282304, 1884241920, 321699840, 146227200, 134529024, 245317632, 1063043072, 49963024384, 105517547520, 15073935360, 5514854400, 3843686400

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers, A068555, based on Patalan numbers of order 8, A025753.

			T(0..4,0..4) is
  1           8           288         13056       652800
  56          224         5376        182784      7311360
  3360        8960        161280      4386816     146227200
  206080      412160      5935104     134529024   3843686400
  12776960    20443136    245317632   4766171136  119154278400

Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.

Cf. A068555, A025753, A034977 (first row), A216704 (first column), A248324, A248325, A248326, A248328, A248329.

matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*64^(n+k)*binomial(n-1/8,n+k)) \\ Michel Marcus, Oct 09 2014

T(0,0)=1, T(n,k) = T(n-1,k)*(64*n-8)/(n+k), T(n,k) = T(n,k-1)*(64*k-56)/(n+k).
G.f.: (x/(1-64*x)^(7/8)+y/(1-64*y)^(1/8))/(x+y-64*x*y).
T(n,k) = (-1)^k*64^(n+k)*binomial(n-1/8,n+k).

A283150 Riordan array (1/(1-9x)^(1/3), x/(9x-1)).

Original entry on oeis.org

1, 3, -1, 18, -12, 1, 126, -126, 21, -1, 945, -1260, 315, -30, 1, 7371, -12285, 4095, -585, 39, -1, 58968, -117936, 49140, -9360, 936, -48, 1, 480168, -1120392, 560196, -133380, 17784, -1368, 57, -1, 3961386, -10563696, 6162156, -1760616, 293436, -30096, 1881, -66, 1, 33011550, -99034650, 66023100

Offset: 0

Tom Richardson, Mar 01 2017

Triangle read by rows. This is an example of a Riordan group involution. Dual Riordan array of A283151. With A283151 and A248324, forms doubly infinite Riordan array. For b and c the sequences A283151 and A248324, respectively, and i,j >= 0, the doubly infinite array with d(i,j) = a(i,j), d(-j,-i) = b(i,j), d(i,-j) = c(j,i), and d(-i,j) = 0 (except d(0,0)=1) is a doubly infinite Riordan array.

Matrix inverse of a(m,n) is a(m,n). - Werner Schulte, Aug 05 2017

			The triangle begins
        1;
        3,        -1;
       18,       -12,       1;
      126,      -126,      21,       -1;
      945,     -1260,     315,      -30,      1;
     7371,    -12285,    4095,     -585,     39,     -1;
    58968,   -117936,   49140,    -9360,    936,    -48,    1;
   480168,  -1120392,  560196,  -133380,  17784,  -1368,   57,  -1;
  3961386, -10563696, 6162156, -1760616, 293436, -30096, 1881, -66, 1;

Peter Bala, A 4-parameter family of embedded Riordan arrays
Peter Bala, A note on the diagonals of a proper Riordan Array
H. Prodinger, Some information about the binomial transform, The Fibonacci Quarterly, 32, 1994, 412-415.
Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.

Cf. A004987, A248324, A283151, A046521, A155579.

T := (n, k) -> (-1)^k*binomial(n - 2/3, n - k)*9^(n - k):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Sep 03 2021

a(m,n) = binomial(-n-1/3, m-n)*(-1)^m*9^(m-n);
tabl(nn) = for(n=0, nn, for (k=0, n, print1(a(n, k), ", ")); print); \\ Michel Marcus, Aug 07 2017

a(m,n) = binomial(-n-1/3, m-n)*(-1)^m*9^(m-n).
G.f.: (1-9x)^(2/3)/(xy-9x+1).
Recurrence: a(m,n) = a(m, n-1)*(n-1-m)/(9*n-6) for 0 < n <= m. - Werner Schulte, Aug 05 2017
From Peter Bala, Mar 05 2018 (Start):
Let P(n,x) = Sum_{k = 0..n} T(n,k)*x^(n-k) denote the n-th row polynomial in descending powers of x. Then (-1)^n*P(n,x) is the n-th degree Taylor polynomial of (1 - 9*x)^(n-2/3) about 0. For example, for n = 4 we have (1 - 9*x)^(10/3) = 945*x^4 - 1260*x^3 + 315*x^2 - 30*x + 1 + O(x^5).
Let R(n,x) denote the n-th row polynomial of this triangle. The polynomial R(n,9*x) has the e.g.f. Sum_{k = 0..n} T(n,k)*(9*x)^k/k!. The e.g.f. for the n-th diagonal of the triangle (starting at n = 0 for the main diagonal) equals exp(-x) * the e.g.f. for the polynomial R(n,9*x). For example, when n = 3 we have exp(-x)*(126 - 126*(9*x) + 21*(9*x)^2/2! - (9*x)^3/3!) = 126 - 1260*x + 4095*x^2/2! - 9360*x^3/3! + 17784*x^4/4! - ....
Let F(x) = (1 - ( 1 - 9*x)^(2/3))/(3*x) denote the o.g.f. of A155579. The derivatives of F(x) are related to the row polynomials P(n,x) by the identity x^n/n! * (d/dx)^n(F(x)) = 1/(3*x)*( (-1)^n - P(n,x)/(1 - 9*x)^(n-2/3) ), n = 0,1,2,.... Cf. A283151 and A046521. (End)
From Peter Bala, Aug 18 2021: (Start)
T(n,k) = (-1)^k*binomial(n-2/3, n-k)*9^(n-k).
Analogous to the binomial transform we have the following sequence transformation formula: g(n) = Sum_{k = 0..n} T(n,k)*b^(n-k)*f(k) iff f(n) = Sum_{k = 0..n} T(n,k)*b^(n-k)*g(k). See Prodinger, bottom of p. 413, with b replaced with 9*b, c = -1 and d = 1/3.
Equivalently, if F(x) = Sum_{n >= 0} f(n)*x^n and G(x) = Sum_{n >= 0} g(n)*x^n are a pair of formal power series then
G(x) = (1/(1 - 9*b*x)^(1/3)) * F(x/(1 - 9*b*x)) iff F(x) = (1/(1 + 9*b*x)^(1/3)) * G(x/(1 + 9*b*x)).
The infinitesimal generator of the unsigned array has the sequence (9*n+3) n>=0 on the main subdiagonal and zeros elsewhere. The m-th power of the unsigned array has entries m^(n-k)*|T(n,k)|. (End)

Offset corrected by Werner Schulte, Aug 05 2017

A283151 Triangle read by rows: Riordan array (1/(1-9x)^(2/3), x/(9x-1)).

Original entry on oeis.org

1, 6, -1, 45, -15, 1, 360, -180, 24, -1, 2970, -1980, 396, -33, 1, 24948, -20790, 5544, -693, 42, -1, 212058, -212058, 70686, -11781, 1071, -51, 1, 1817640, -2120580, 848232, -176715, 21420, -1530, 60, -1, 15677145, -20902860, 9754668, -2438667, 369495, -35190, 2070, -69, 1, 135868590, -203802885

Offset: 0

Tom Richardson, Mar 01 2017

This is an example of a Riordan group involution.
Dual Riordan array of A283150.
With A283150 and A248324, forms doubly infinite Riordan array. For b and c the sequences A283150 and A248324, respectively, and i,j >= 0, the doubly infinite array with d(i,j) = a(i,j), d(-j,-i) = b(i,j), d(i,-j) = c(i,j), and d(-i,j) = 0 (except d(0,0)=1) is a doubly infinite Riordan array.

			Triangle begins
         1;
         6,        -1;
        45,       -15,       1;
       360,      -180,      24,       -1;
      2970,     -1980,     396,      -33,      1;
     24948,    -20790,    5544,     -693,     42,     -1;
    212058,   -212058,   70686,   -11781,   1071,    -51,    1;
   1817640,  -2120580,  848232,  -176715,  21420,  -1530,   60,  -1;
  15677145, -20902860, 9754668, -2438667, 369495, -35190, 2070, -69, 1;

Peter Bala, A 4-parameter family of embedded Riordan arrays
Peter Bala, A note on the diagonals of a proper Riordan Array
H. Prodinger, Some information about the binomial transform, The Fibonacci Quarterly, 32, 1994, 412-415.
Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.

Cf. A004988, A248324, A283150, A025748, A046521.

a(m,n) = binomial(-n-2/3, m-n)*(-1)^m*9^(m-n).
G.f.: (1-9x)^(1/3)/(xy-9x+1).
Recurrence: a(m,n) = a(m,n-1)*(n-1-m)/(9*n-3) for 0 < n <= m; matrix inverse of a(m,n) is a(m,n). - Werner Schulte, Aug 05 2017
From Peter Bala, Mar 05 2018 (Start):
Let P(n,x) = Sum_{k = 0..n} T(n,k)*x^(n-k) denote the n-th row polynomial in descending powers of x. Then (-1)^n*P(n,x) is the n-th degree Taylor polynomial of (1 - 9*x)^(n-1/3) about 0. For example, for n = 4 we have (1 - 9*x)^(11/3) = 2970*x^4 - 1980*x^3 + 396*x^2 - 33*x + 1 + O(x^5).
Let R(n,x) denote the n-th row polynomial of this triangle. The polynomial R(n,9*x) has the e.g.f. Sum_{k = 0..n} T(n,k)*(9*x)^k/k!. The e.g.f. for the n-th diagonal of the triangle (starting at n = 0 for the main diagonal) equals exp(-x) * the e.g.f. for the polynomial R(n,9*x). For example, when n = 3 we have exp(-x)*(360 - 180*(9*x) + 24*(9*x)^2/2! - (9*x)^3/3!) = 360 - 1980*x + 5544*x^2/2! - 11781*x^3/3! + 21420*x^4/4! - ....
Let F(x) = (1 - ( 1 - 9*x)^(1/3))/(3*x). See A025748. The derivatives of F(x) are related to the row polynomials P(n,x) by the identity x^n/n! * (d/dx)^n(F(x)) = 1/(3*x)*( (-1)^n - P(n,x)/(1 - 9*x)^(n-1/3) ), n = 0,1,2,.... Cf. A283151 and A046521. (End)
From Peter Bala, Aug 18 2021: (Start)
T(n,k) = (-1)^k*binomial(n-1/3, n-k)*9^(n-k).
Analogous to the binomial transform we have the following sequence transformation formula: g(n) = Sum_{k = 0..n} T(n,k)*b^(n-k)*f(k) iff f(n) = Sum_{k = 0..n} T(n,k)*b^(n-k)*g(k). See Prodinger, bottom of p. 413, with b replaced with 9*b, c = -1 and d = 2/3.
Equivalently, if F(x) = Sum_{n >= 0} f(n)*x^n and G(x) = Sum_{n >= 0} g(n)*x^n are a pair of formal power series then
G(x) = (1/(1 - 9*b*x)^(2/3)) * F(x/(1 - 9*b*x)) iff F(x) = (1/(1 + 9*b*x)^(2/3)) * G(x/(1 + 9*b*x)).
The infinitesimal generator of the unsigned array has the sequence (9*n+6) n>=0 on the main subdiagonal and zeros elsewhere. The m-th power of the unsigned array has entries m^(n-k)*|T(n,k)|. (End)

Offset corrected by Werner Schulte, Aug 05 2017

cp's OEIS Frontend

A248325 Square array read by antidiagonals downwards: super Patalan numbers of order 4.

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A248326 Square array read by downward antidiagonals: super Patalan numbers of order 5.

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A248328 Square array read by antidiagonals downwards: super Patalan numbers of order 6.

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A248329 Square array read by antidiagonals downwards: super Patalan numbers of order 7.

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A248332 Square array read by antidiagonals downwards: super Patalan numbers of order 8.

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A283150 Riordan array (1/(1-9x)^(1/3), x/(9x-1)).

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A283151 Triangle read by rows: Riordan array (1/(1-9x)^(2/3), x/(9x-1)).

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