Real-rooted atlas

A371081 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-54 + 21\,x + 2\,x^{2} - x^{3} - 81\,n + 31\,n x - 2\,n x^{2} - 45\,n^{2} + 14\,n^{2} x - n^{2} x^{2} - 11\,n^{3} + 2\,n^{3} x - n^{4}\right)\,P_{n-2}(x) + \left(-42\,x + 29\,x^{2} - 5\,x^{3} - 47\,n x + 22\,n x^{2} - 2\,n x^{3} - 17\,n^{2} x + 4\,n^{2} x^{2} - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-6\,x^{2} + 5\,x^{3} - x^{4} - 5\,n x^{2} + 2\,n x^{3} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(22 - 6\,x + x^{2} + 13\,n - 2\,n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(9\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 17 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
117118
2452541507
3173483180119120648
49086722277601299222011149645
56231283220029376155291239920365183935406
654184320002142778688212024320737587210210056217780713543
7582473664000275245327872333887637121476251840275674122292928191892611804948
87582898073600041898359098368605823186508832717225113677401499528665136046704011441124120571220089
91175286379622400074671073087201281260392761910016808776405168642329814281920329626983042387256328814721545120563604523959882
102137852808146944000154122712525486080029881294633584230422323959818676224761890146814464130929417610241196125800325926874241564740203014000991942851733043
114508579355769036800003648695693320101888008021049596763877785668679465988529479682721701926449034245527880549901824612200745381763821000656001352386244264110026071903083707152130033600
121090862527938940108800009822631349544648376320024225957884527275884544234775881183119295283210637702708055632281625029190426782740483269636545989964824700636154854410999474428802877547024427363232841233995195876296848792453
1330005916166124969656320000298404908023434578362368008184269890819748854185984888523222761341189406720454682974927980932628481220486171187439841280184340868523310563841640051276545011208829008297998722903410699904577266747266716004069168965907464691293074654054
14930847387538806150791168000010159973250529009393454284800307564987356291100191621120037086382436231912255284838421219971896422709682651136641980282045416169648128110411534927185803837441133942422258580090887179819327554897922851473777572160712139197369611012599104100984524970122936833931289344635073454655
Technical details
Node id
n-abfce20d05838fa2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f44590ad6843dfc1

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:                 1;
  1:                17,                1;
  2:               452,               54,                1;
  3:             17348,             3180,              119,              1;
  4:            908672,           227760,            12992,            220,             1;
  5:          62312832,         20029376,          1552912,          39920,           365,           1;
  6:        5418432000,       2142778688,        212024320,        7375872,        102100,         562,         1;
  7:      582473664000,     275245327872,      33388763712,     1476251840,      27567412,      229292,       819,      1;
  8:    75828980736000,   41898359098368,    6058231865088,   327172251136,    7740149952,    86651360,    467040,   1144,    1;
  9: 11752863796224000, 7467107308720128, 1260392761910016, 80877640516864, 2329814281920, 32962698304, 238725632, 881472, 1545, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4+11*n^3+45*n^2+81*n+54-(2*n^3+14*n^2+31*n+21)*x+(n^2+2*n-2)*x^2+x^3)*P(n-2,x) - ((2*n^3+17*n^2+47*n+42)*x-(4*n^2+22*n+29)*x^2+(2*n+5)*x^3)*d/dx P(n-2,x) - ((n^2+5*n+6)*x^2-(2*n+5)*x^3+x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+13*n+22-(2*n+6)*x+x^2)*P(n-1,x) + (2*n+9)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 17 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 17 + x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 11*n^3 + 45*n^2 + 81*n + 54 - (2*n^3 + 14*n^2 + 31*n + 21)*x + (n^2 + 2*n - 2)*x^2 + x^3)*P[n - 2, x] - ((2*n^3 + 17*n^2 + 47*n + 42)*x - (4*n^2 + 22*n + 29)*x^2 + (2*n + 5)*x^3)*D[P[n - 2, x], x] - ((n^2 + 5*n + 6)*x^2 - (2*n + 5)*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 13*n + 22 - (2*n + 6)*x + x^2)*P[n - 1, x] + (2*n + 9)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 17 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 11*n^3 + 45*n^2 + 81*n + 54 - (2*n^3 + 14*n^2 + 31*n + 21)*x + (n^2 + 2*n - 2)*x^2 + x^3)*v[i - 2] - ((2*n^3 + 17*n^2 + 47*n + 42)*x - (4*n^2 + 22*n + 29)*x^2 + (2*n + 5)*x^3)*deriv(v[i - 2], x) - ((n^2 + 5*n + 6)*x^2 - (2*n + 5)*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 13*n + 22 - (2*n + 6)*x + x^2)*v[i - 1] + (2*n + 9)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.