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 1-10 of 31 results. Next

A223168 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)*(x^(1/2)*d/dx)^n when n is odd, and of 2^(n/2)*(x^(1/2)*d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 12, 4, 15, 20, 4, 15, 90, 60, 8, 105, 210, 84, 8, 105, 840, 840, 224, 16, 945, 2520, 1512, 288, 16, 945, 9450, 12600, 5040, 720, 32, 10395, 34650, 27720, 7920, 880, 32, 10395, 124740, 207900, 110880, 23760, 2112, 64, 135135, 540540, 540540, 205920, 34320, 2496, 64
Offset: 0

Views

Author

Udita Katugampola, Mar 17 2013

Keywords

Comments

Also coefficients in the expansion of k-th derivative of exp(n*x^2), see Mathematica program. - Vaclav Kotesovec, Jul 16 2013

Examples

			Triangle begins:
       1;
       1,      2;
       3,      2;
       3,     12,      4;
      15,     20,      4;
      15,     90,     60,      8;
     105,    210,     84,      8;
     105,    840,    840,    224,    16;
     945,   2520,   1512,    288,    16;
     945,   9450,  12600,   5040,   720,   32;
   10395,  34650,  27720,   7920,   880,   32;
   10395, 124740, 207900, 110880, 23760, 2112, 64;
  135135, 540540, 540540, 205920, 34320, 2496, 64;
  .
Expansion takes the form:
2^0 (x^(1/2)*d/dx)^1 = 1*x^(1/2)*d/dx.
2^1 (x^(1/2)*d/dx)^2 = 1*d/dx + 2*x*d^2/dx^2.
2^1 (x^(1/2)*d/dx)^3 = 3*x^(1/2)*d^2/dx^2 + 2*x^(3/2)*d^3/dx^3.
2^2 (x^(1/2)*d/dx)^4 = 3*d^2/dx^2 + 12*x*d^3/dx^3 + 4*x^2*d^4/dx^4.
2^2 (x^(1/2)*d/dx)^5 = 15*x^(1/2)*d^3/dx^3 + 20*x^(3/2)*d^4/dx^4 + 4*x^(5/2)*d^5/dx^5.
`
`

Links

Crossrefs

Odd rows includes absolute values of A098503 from right to left.
Cf. A223169-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

Programs

  • Maple
    a[0]:= f(x);
for i from 1 to 13 do
a[i]:= simplify(2^((i+1)mod 2)*x^(1/2)*(diff(a[i-1],x$1)));
end do;
  • Mathematica
    Flatten[CoefficientList[Expand[FullSimplify[Table[D[E^(n*x^2),{x,k}]/(E^(n*x^2)*(2*n)^Floor[(k+1)/2]),{k,1,13}]]]/.x->1,n]] (* Vaclav Kotesovec, Jul 16 2013 *)

A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)*(x^(5/6)*d/dx)^n when n=0,2,4,6,...

Original entry on oeis.org

1, 1, 6, 7, 84, 36, 91, 1638, 1404, 216, 1729, 41496, 53352, 16416, 1296, 43225, 1296750, 2223000, 1026000, 162000, 7776, 1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656, 49579075, 2082321150, 5354540100, 4118877000, 1300698000, 187300512
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			Triangle begins:
1;
1, 6;
7, 84, 36;
91, 1638, 1404, 216;
1729, 41496, 53352, 16416, 1296;
43225, 1296750, 2223000, 1026000, 162000, 7776;
1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656;
49579075, 2082321150, 5354540100, 4118877000, 1300698000, 187300512, 12083904, 279936;

Links

Crossrefs

Even rows of A223172.
Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j];
end do;

A223511 Triangle T(n,k) represents the coefficients of (x^9*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 9, 1, 153, 27, 1, 3825, 855, 54, 1, 126225, 32895, 2745, 90, 1, 5175225, 1507815, 150930, 6705, 135, 1, 253586025, 80565975, 9205245, 499590, 13860, 189, 1, 14454403425, 4926412575, 623675430, 39180645, 1345050, 25578, 252, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Comments

Also the Bell transform of A045755(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

Examples

			1;
9,1;
153,27,1;
3825,855,54,1;
126225,32895,2745,90,1;
5175225,1507815,150930,6705,135,1;
253586025,80565975,9205245,499590,13860,189,1;
14454403425,4926412575,623675430,39180645,1345050,25578,252,1;

Crossrefs

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223512-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    b[0]:=g(x):
for j from 1 to 10 do
b[j]:=simplify(x^9*diff(b[j-1],x$1);
end do;
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(8*k+1, k=0..n), 10); # Peter Luschny, Jan 29 2016
  • Mathematica
    rows = 8;
t = Table[Product[8k+1, {k, 0, n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

A223523 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)*(x^(1/2)*d/dx)^n, where n = 1, 3, 5, ...

Original entry on oeis.org

1, 3, 2, 15, 20, 4, 105, 210, 84, 8, 945, 2520, 1512, 288, 16, 10395, 34650, 27720, 7920, 880, 32, 135135, 540540, 540540, 205920, 34320, 2496, 64, 2027025, 9459450, 11351340, 5405400, 1201200, 131040, 6720, 128
Offset: 1

Views

Author

Udita Katugampola, Mar 21 2013

Keywords

Comments

Triangle S(n,n-k) by rows: coefficients of 2^n * |L(n,1/2,x)|, with L the generalized Laguerre polynomials.

Examples

			Triangle begins:
1;
3, 2;
15, 20, 4;
105, 210, 84, 8;
945, 2520, 1512, 288, 16;
10395, 34650, 27720, 7920, 880, 32;
135135, 540540, 540540, 205920, 34320, 2496, 64;
.
.
Expansion takes the form:
2^0 (x^(1/2)*d/dx)^1 = 1*x^(1/2)*d/dx.
2^1 (x^(1/2)*d/dx)^3 = 3*x^(1/2)*d^2/dx^2 + 2*x^(3/2)*d^3/dx^3.
2^2 (x^(1/2)*d/dx)^5 = 15*x^(1/2)*d^3/dx^3 + 20*x^(3/2)*d^4/dx^4 + 4*x^(5/2)*d^5/dx^5.

Links

Crossrefs

Cf. A223168-A223172.
Rows includes odd rows of A223168.
Rows includes absolute values of A098503 from right to left of the triangular form.

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 20 do
a[i]:= simplify(2^((i+1)mod 2)*x^(1/2)*(diff(a[i-1],x$1)));
end do:
for j from 1 to 10 do
b[j]:=a[2j-1];
end do;

Formula

T(n, k) = 2^n * n!/(n-k)! * C(n+1/2, k), n>=0, k<=n.

A223522 Triangle T(n,k) represents the coefficients of (x^20*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 20, 1, 780, 60, 1, 45240, 4320, 120, 1, 3483480, 382200, 13800, 200, 1, 334414080, 40556880, 1734600, 33600, 300, 1, 38457619200, 5039012160, 243505080, 5699400, 69300, 420, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			1;
20,1;
780,60,1;
45240,4320,120,1;
3483480,382200,13800,200,1;
334414080,40556880,1734600,33600,300,1;
38457619200,5039012160,243505080,5699400,69300,420,1;
5153320972800,718724260800,38155703040,1024322880,15262800,127680,560,1;

Crossrefs

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^20*diff(b[j-1],x$1);
end do;

A098503 Triangle T(n,k) by rows: coefficient [x^(n-k)] of 2^n * n! *L(n,1/2,x), with L the generalized Laguerre polynomials in the Abramowitz-Stegun normalization.

Original entry on oeis.org

1, -2, 3, 4, -20, 15, -8, 84, -210, 105, 16, -288, 1512, -2520, 945, -32, 880, -7920, 27720, -34650, 10395, 64, -2496, 34320, -205920, 540540, -540540, 135135, -128, 6720, -131040, 1201200, -5405400, 11351340, -9459450, 2027025, 256, -17408
Offset: 0

Views

Author

Ralf Stephan, Sep 15 2004

Keywords

Examples

			2^0 *0! *L(0,1/2,x) = 1.
2^1 *1! *L(1,1/2,x) = -2*x + 3.
2^2 *2! *L(2,1/2,x) = 4*x^2 - 20*x + 15.
2^3 *3! *L(3,1/2,x) = -8*x^3 + 84*x^2 - 210*x + 105.
2^4 *4! *L(4,1/2,x) = 16*x^4 - 288*x^3 + 1512*x^2 - 2520*x + 945.
Triangle begins:
    1;
   -2,     3;
    4,   -20,    15;
   -8,    84,  -210,     105;
   16,  -288,  1512,   -2520,    945;
  -32,   880, -7920,   27720, -34650,   10395;
   64, -2496, 34320, -205920, 540540, -540540, 135135;

Links

Crossrefs

Columns include (-1)^n times A000079, n/2*A014480. Diagonals include A001147, -A000906, 4*A001881.
Cf. A223168..A223172 and A223523..A223532.

Programs

  • Mathematica
    Table[Reverse[Table[2^n*(-1)^k*n!/k!*Binomial[n + 1/2, n - k], {k, 0, n}]], {n, 0, 7}] (* T. D. Noe, Apr 05 2013 *)

Formula

T(n, k) = (-2)^n * (-1)^k * n!/(n-k)! * binomial(n+1/2,k), = (-1)^(n+k) *2^(n-2k) *k! *binomial(2n+1,2k)*binomial(2k,k), n>=0, k<=n.

A223536 Coefficients of (x^(1/6)*d/dx)^n for positive integer n.

Original entry on oeis.org

1, 1, 6, -2, 9, 8, 6, 13, 36, 36, -42, 70, -75, 180, 108, 798, -1162, 945, -630, 1620, 648, 3192, -4284, 3052, -1575, 630, -2268, -648, 92568, -117684, 77588, -35637, 12600, -1512, 18144, 3888, 1573656
Offset: 1

Views

Author

Udita Katugampola, Apr 18 2013

Keywords

Comments

These are generalized Stirling numbers.

Examples

			1;
1, 6;
-2, 9, 8;
6, 13, 36, 36;
-42, 70, -75, 180, 108;
798, -1162, 945, -630, 1620, 648;

Links

Crossrefs

Cf. A223168-A223172, A223533-A223536.

Programs

  • Maple
    # This will generate the sequence as coefficients of pseudo polynomials
# up to a constant multiple.
a[0] := f(x):
for i to 10 do
a[i] := simplify(x^(1/6)*(diff(a[i-1],x$1)))
end do;

Formula

G.f.: exp(((1+5/6*x*y)^(6/5)-1)/x).

A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)*(x^(1/3)*d/dx)^n when n is odd, and of 3^(n/2)*(x^(2/3)*d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 3, 4, 3, 4, 24, 9, 28, 42, 9, 28, 252, 189, 27, 280, 630, 270, 27, 280, 3360, 3780, 1080, 81, 3640, 10920, 7020, 1404, 81, 3640, 54600, 81900, 35100, 5265, 243, 58240, 218400, 187200, 56160, 6480, 243, 58240, 1048320, 1965600
Offset: 0

Views

Author

Udita Katugampola, Mar 18 2013

Keywords

Examples

			Triangle begins:
1;
1, 3;
4, 3;
4, 24, 9;
28, 42, 9;
28, 252, 189, 27;
280, 630, 270, 27;
280, 3360, 3780, 1080, 81;
3640, 10920, 7020, 1404, 81;
3640, 54600, 81900, 35100, 5265, 243,
58240, 218400, 187200, 56160, 6480, 243

Links

Crossrefs

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(3^((i+1)mod 2)*x^(((i+1)mod 2+1)/3)*(diff(a[i-1],x$1 )));
end do;

A223533 Coefficients of (x^(1/3)*d/dx)^n for positive integer n.

Original entry on oeis.org

1, 1, 3, -1, 9, 9, 1, -1, 18, 9, -5, 5, 15, 90, 27, 35, -35, 225, 405, 81, -105, 105, -35, 630, 567, 81, 1155, -1155, 490, -105, 4158, 2268, 243, 15015, -15015, 6895, 945, -10206, -23814, -8748, -729, 75075, -75075, 35700, -10675, 2835, -945, 34020, 41310, 10935, 729
Offset: 1

Views

Author

Udita Katugampola, Apr 18 2013

Keywords

Comments

These are generalized Stirling numbers.

Examples

			1;
1, 3;
-1, 9, 9;
1, -1, 18, 9;
-5, 5, 15, 90, 27;
35, -35, 225, 405, 81;
-105, 105, 630, 567, -35, 81;
1155, -1155, 630, 4158, 490, 2268, -105, 243;

Links

Crossrefs

Cf. A223168-A223172, A223534-A223536.

Programs

  • Maple
    # This will generate the sequence as coefficients of pseudo polynomials
# up to a constant multiple.
a[0] := f(x):
for i to 10 do
a[i] := simplify(x^(1/3)*(diff(a[i-1],x$1)))
end do;

Formula

G.f.: exp(((1+2/3*x*y)^(3/2)-1)/x).

A223170 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)*(x^(1/4)*d/dx)^n when n is odd, and of 4^(n/2)*(x^(3/4)*d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 4, 5, 4, 5, 40, 16, 45, 72, 16, 45, 540, 432, 64, 585, 1404, 624, 64, 585, 9360, 11232, 3328, 256, 9945, 31824, 21216, 4352, 256, 9945, 198900, 318240, 141440, 21760, 1024, 208845, 835380, 742560, 228480, 26880, 1024, 208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096
Offset: 0

Views

Author

Udita Katugampola, Mar 20 2013

Keywords

Examples

			Triangle begins:
1;
1, 4;
5, 4;
5, 40, 16;
45, 72, 16;
45, 540, 432, 64;
585, 1404, 624, 64;
585, 9360, 11232, 3328, 256;
9945, 31824, 21216, 4352, 256;
9945, 198900, 318240, 141440, 21760, 1024;
208845, 835380, 742560, 228480, 26880, 1024;
208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096;

Links

Crossrefs

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(4^((i+1)mod 2)*x^((2((i+1)mod 2)+1)/4)*(diff(a[i-1],x$1 )));
end do;
  • Mathematica
    nmax = 12;
b[0] = Exp[x]; For[ i = 1 , i <= nmax , i++, b[i] = 4^Mod[i + 1, 2]*x^((2 Mod[i + 1, 2] + 1)/4)*D[b[i - 1], x]] // Simplify;
row[1] = {1}; row[n_] := List @@ Expand[b[n]/f[x]] /. x -> 1;
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Feb 22 2019, from Maple *)

Extensions

Missing terms inserted by Jean-François Alcover, Feb 22 2019
Showing 1-10 of 31 results. Next

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