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 15 results. Next

A128577 Self-convolution of A128318.

Original entry on oeis.org

1, 2, 9, 64, 624, 7736, 116416, 2060808, 41952600, 965497440, 24786054816, 702201877920, 21761251764672, 732269872931712, 26589359234860560, 1036241806935453696, 43142510740036313088, 1911022260200150482944, 89737455913330610995200, 4452805047268938247981056, 232806644343118618035904512, 12791828071344703747110764544, 736928909474399720669590216704, 44416721474748725213260027514880
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Comments

A128318 equals row 0 of table A128570.

Links

Crossrefs

Cf. A128570 (triangle), rows: A128318, A128571, A128572, A128573, A128574, A128575, A128576; A128578 (main diagonal), A128579 (antidiagonal sums).

Programs

  • PARI
    {a(n)=local(A=1+(n+1)*x);for(j=0,n,A=1+(n+1-j)*x*A^2 +x*O(x^n)); polcoeff(A^2,n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f.: A(x) = [1 + x*R(x,1)^2]^2, where R(x,1) = 1 + 2*x*R(x,2)^2, R(x,2) = 1 + 3*x*R(x,3)^2, ..., R(x,n) = 1 + (n+1)*x*R(x,n+1)^2, ... and R(x,n) is the g.f. of row n of table A128570.
a(n) ~ 2*A128318(n). - Vaclav Kotesovec, Mar 19 2016

A128570 Rectangular table, read by antidiagonals, where the g.f. of row n, R(x,n), satisfies: R(x,n) = 1 + (n+1)*x*R(x,n+1)^2 for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 12, 28, 1, 4, 24, 114, 276, 1, 5, 40, 288, 1440, 3480, 1, 6, 60, 580, 4440, 22368, 53232, 1, 7, 84, 1020, 10560, 82080, 409248, 955524, 1, 8, 112, 1638, 21420, 226560, 1752000, 8585088, 19672320, 1, 9, 144, 2464, 38976, 523320, 5532960, 42178800, 202733760, 456803328, 1, 10, 180, 3528, 65520, 1068480, 14399280, 150570240, 1127335680, 5317663680, 11810032896, 1, 11, 220, 4860, 103680, 1991808, 32716992, 437433780, 4501422240, 33073099200, 153345634560, 336463895808
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Comments

Row r > 0 is asymptotic to 2^(2*r) * n^r * A128318(n) / (3^r * r!). - Vaclav Kotesovec, Mar 19 2016

Examples

			Row g.f.s satisfy: R(x,0) = 1 + x*R(x,1)^2, R(x,1) = 1 + 2x*R(x,2)^2,
R(x,2) = 1 + 3x*R(x,3)^2, R(x,3) = 1 + 4x*R(x,4)^2, ...
where the initial rows begin:
R(x,0):[1,1,4,28,276,3480,53232,955524,19672320,456803328,...];
R(x,1):[1,2,12,114,1440,22368,409248,8585088,202733760,...];
R(x,2):[1,3,24,288,4440,82080,1752000,42178800,1127335680,...];
R(x,3):[1,4,40,580,10560,226560,5532960,150570240,4501422240,...];
R(x,4):[1,5,60,1020,21420,523320,14399280,437433780,14479664640,...];
R(x,5):[1,6,84,1638,38976,1068480,32716992,1098069504,39896236800,...];
R(x,6):[1,7,112,2464,65520,1991808,67189248,2469837888,97765355520,..];
R(x,7):[1,8,144,3528,103680,3461760,127569600,5098406400,...];
R(x,8):[1,9,180,4860,156420,5690520,227470320,9821970180,...];
R(x,9):[1,10,220,6490,227040,8939040,385265760,17875608960,..].

Links

Crossrefs

Rows: A128318, A128571, A128572, A128573, A128574, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).
Cf A268652.

Programs

  • PARI
    {T(n,k)=local(A=1+(n+k+1)*x); for(j=0,k,A=1+(n+k+1-j)*x*A^2 +x*O(x^k));polcoeff(A,k)}
for(n=0, 12, for(k=0, 10, print1(T(n, k), ", ")); print(""))

A128571 Row 1 of table A128570.

Original entry on oeis.org

1, 2, 12, 114, 1440, 22368, 409248, 8585088, 202733760, 5317663680, 153345634560, 4821848409600, 164211751261440, 6022162697840640, 236652023784960000, 9921992082873223680, 442138176056374548480, 20869300232695599552000, 1040210006521640127367680, 54600929159270409876879360
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Links

Crossrefs

Cf. A128570 (triangle), other rows: A128318, A128572, A128573, A128574, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).
Cf. A268652.

Programs

  • PARI
    {a(n)=local(A=1+(n+2)*x);for(j=0,n,A=1+(n+2-j)*x*A^2 +x*O(x^n)); polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f.: A(x) = 1 + 2x*R(x,2)^2, where R(x,2) = 1 + 3*x*R(x,3)^2, R(x,3) = 1 + 4*x*R(x,4)^2, ..., R(x,n) = 1 + (n+1)*x*R(x,n+1)^2, ... and R(x,n) is the g.f. of row n of table A128570.
a(n) ~ 4*n*A128318(n)/3. - Vaclav Kotesovec, Mar 19 2016

A128572 Row 2 of table A128570.

Original entry on oeis.org

1, 3, 24, 288, 4440, 82080, 1752000, 42178800, 1127335680, 33073099200, 1055891810880, 36435757294080, 1351364788224000, 53617083034314240, 2266453101278568960, 101705245560225146880, 4829501671573344393600
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Links

Crossrefs

Cf. A128570 (triangle), other rows: A128318, A128571, A128573, A128574, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).

Programs

  • PARI
    {a(n)=local(A=1+(n+3)*x);for(j=0,n,A=1+(n+3-j)*x*A^2 +x*O(x^n)); polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + 3x*R(x,3)^2, where R(x,3) = 1 + 4*x*R(x,4)^2, R(x,4) = 1 + 5*x*R(x,5)^2, ..., R(x,n) = 1 + (n+1)*x*R(x,n+1)^2, ... and R(x,n) is the g.f. of row n of table A128570.
a(n) ~ 8*n^2*A128318(n)/9. - Vaclav Kotesovec, Mar 19 2016

A128573 Row 3 of table A128570.

Original entry on oeis.org

1, 4, 40, 580, 10560, 226560, 5532960, 150570240, 4501422240, 146351879520, 5135738294400, 193376042294400, 7775407679034240, 332528365742227200, 15073953619379719680, 722117116504240994880, 36458486578829035929600
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Links

Crossrefs

Cf. A128570 (triangle), other rows: A128318, A128571, A128572, A128574, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).

Programs

  • PARI
    {a(n)=local(A=1+(n+4)*x);for(j=0,n,A=1+(n+4-j)*x*A^2 +x*O(x^n)); polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + 4x*R(x,4)^2, where R(x,4) = 1 + 5*x*R(x,5)^2, R(x,5) = 1 + 6*x*R(x,6)^2, ..., R(x,n) = 1 + (n+1)*x*R(x,n+1)^2, ... and R(x,n) is the g.f. of row n of table A128570.
a(n) ~ 32*n^3*A128318(n)/81. - Vaclav Kotesovec, Mar 19 2016

A128574 Row 4 of table A128570.

Original entry on oeis.org

1, 5, 60, 1020, 21420, 523320, 14399280, 437433780, 14479664640, 517426156800, 19824547680000, 810083131361280, 35155640625638400, 1614680474921256960, 78256021787814850560, 3991780109967777792000, 213813097136418588641280
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Links

Crossrefs

Cf. A128570 (triangle), other rows: A128318, A128571, A128572, A128573, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).

Programs

  • PARI
    {a(n)=local(A=1+(n+5)*x);for(j=0,n,A=1+(n+5-j)*x*A^2 +x*O(x^n)); polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + 5x*R(x,5)^2, where R(x,5) = 1 + 6*x*R(x,6)^2, R(x,6) = 1 + 7*x*R(x,7)^2, ..., R(x,n) = 1 + (n+1)*x*R(x,n+1)^2, ... and R(x,n) is the g.f. of row n of table A128570.
a(n) ~ 32*n^4*A128318(n)/243. - Vaclav Kotesovec, Mar 19 2016

A128575 Row 5 of table A128570.

Original entry on oeis.org

1, 6, 84, 1638, 38976, 1068480, 32716992, 1098069504, 39896236800, 1555603999488, 64678765165056, 2853714891138048, 133101200708356608, 6542154022577467392, 337978986519657627648, 18310837206705702672384
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Links

Crossrefs

Cf. A128570 (triangle), other rows: A128318, A128571, A128572, A128573, A128574, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).

Programs

  • PARI
    {a(n)=local(A=1+(n+6)*x);for(j=0,n,A=1+(n+6-j)*x*A^2 +x*O(x^n)); polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + 6x*R(x,6)^2, where R(x,6) = 1 + 7*x*R(x,7)^2, R(x,7) = 1 + 8*x*R(x,8)^2, ..., R(x,n) = 1 + (n+1)*x*R(x,n+1)^2, ... and R(x,n) is the g.f. of row n of table A128570.
a(n) ~ 128*n^5*A128318(n)/3645. - Vaclav Kotesovec, Mar 19 2016

A128576 Row 6 of table A128570.

Original entry on oeis.org

1, 7, 112, 2464, 65520, 1991808, 67189248, 2469837888, 97765355520, 4132860197760, 185458263419520, 8794132843507200, 439083652465543680, 23017956568726049280, 1263929372436815078400, 72550400791147384412160
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Comments

In general, row r > 0 of A128570 is asymptotic to 2^(2*r) * n^r * A128318(n) / (3^r * r!). - Vaclav Kotesovec, Mar 19 2016

Links

Crossrefs

Cf. A128570 (triangle), other rows: A128318, A128571, A128572, A128573, A128574, A128575; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).

Programs

  • PARI
    {a(n)=local(A=1+(n+7)*x);for(j=0,n,A=1+(n+7-j)*x*A^2 +x*O(x^n)); polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + 7x*R(x,7)^2, where R(x,7) = 1 + 8*x*R(x,8)^2, R(x,8) = 1 + 9*x*R(x,9)^2, ..., R(x,n) = 1 + (n+1)*x*R(x,n+1)^2, ... and R(x,n) is the g.f. of row n of table A128570.
a(n) ~ 256*n^6*A128318(n)/32805. - Vaclav Kotesovec, Mar 19 2016

A128578 Main diagonal of table A128570.

Original entry on oeis.org

1, 2, 24, 580, 21420, 1068480, 67189248, 5098406400, 453030209280, 46120247659200, 5290918350734016, 675157532791996800, 94836990558591590400, 14538639675855504384000, 2415072877848471727687680
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Comments

Limit n->infinity (a(n)/n!)^(1/n) = 12.67567... . - Vaclav Kotesovec, Mar 19 2016

Links

Crossrefs

Cf. A128570 (triangle), rows: A128318, A128571, A128572, A128573, A128574, A128575, A128576; A128577 (square of row 0), A128579 (antidiagonal sums).

Programs

  • PARI
    {a(n)=local(A=1+(2*n+1)*x); for(j=0,n,A=1+(2*n+1-j)*x*A^2 +x*O(x^n));polcoeff(A,n)}

A128579 Antidiagonal sums of table A128570.

Original entry on oeis.org

1, 2, 7, 44, 419, 5254, 80687, 1458524, 30259147, 707813762, 18421139495, 527856303160, 16513700403347, 560082210938174, 20471657576850655, 802275966701866964, 33560323690860843995, 1492638035099491033402
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Links

Crossrefs

Cf. A128570 (triangle), rows: A128318, A128571, A128572, A128573, A128574, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal).

Programs

  • PARI
    {a(n)=local(F=1+x,A=0);for(k=0,n, for(j=0,k,F=1+(n+1-j)*x*F^2 +x*O(x^k));A+=polcoeff(F,k));A}

Formula

a(n) ~ exp(1/2) * A128318(n). - Vaclav Kotesovec, Mar 19 2016
Showing 1-10 of 15 results. Next

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