Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A008288 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(31 + 13\,x + x^{2} - 42\,n - 8\,n x + 16\,n^{2}\right)\,P_{n-1}(x) + \left(64 + 78\,x + 19\,x^{2} + x^{3} - 32\,n - 32\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(16 + 32\,x + 18\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
155111
2295840111139
31695076013571111711763
49853940663261063346111221823122363
557412870562633783026191832258111802539361291283667
6334612007665382728510058817636294483167581130012831548545403513598219
719502513651754341905830415510653725968387564213853150845114803530872560259827375541145642179
81136689909351233464240751132201149671401270857761048278686574129031643590117169203323376713948113816130121006471578955451
9662510959625981249215921642205132114234246014881352361470006144112482323467526603832014871012005962235503348821240014650921970861928712935317343852147
10386139653861396501806888568525701708710657843653159929380321841722418016654628702120025877386956566332325811522412344489103774890229261370818044622274825931946527314161659193154256107
112250586812475645491128216902814156931062594620084363160746681909211530197849220871084566185965604170127614049102718550539943331617983412730464098400167205010300993322154924433621401348070074916453730919756511181629920803
12131173812115740857452893912914803195647170548069792938141531204265832226722041901426856723613052588821260943205473263368813536063871207439342063803420304155201316269560604234033133011343771489251647746345820406067630216795012672561449488237071114988571946011
137645370045993898105856141065917692399788571978665807496079413957109913670229698837568603044388278557333074944181351298265793870832252580662942314335335857764771817624076035230996557801364070232330447454209817124004526811288991331515625163447905858506119116418125594141034656640672801771190124915614931
14445604821496238467500864166439039664176686726356015342578776899912262617024824822208342873948232568839170310339385545915583739793497667929833913736736494824548755469589715166337400516780223369243524363935709104501416459491845347252783003271348041239114327649790600674838502511169075690916864626561998538781909745314333518126232688312411669615210059
Technical details
Node id
n-5a23f4110f0a38ce · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-806e832afcb34c9a

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:       5,        5,         1;
  2:      29,       58,        40,        11,          1;
  3:     169,      507,       601,       357,        111,         17,          1;
  4:     985,     3940,      6632,      6106,       3346,       1112,        218,         23,         1;
  5:    5741,    28705,     62633,     78302,      61918,      32258,      11180,       2539,       361,        29,         1;
  6:   33461,   200766,    538272,    851005,     881763,     629448,     316758,     113001,     28315,      4854,       540,       35,       1;
  7:  195025,  1365175,   4341905,   8304155,   10653725,    9683875,    6421385,    3150845,   1148035,    308725,     60259,     8273,     755,      41,      1;
  8: 1136689,  9093512,  33464240,  75113220,  114967140,  127085776,  104827868,   65741290,  31643590,  11716920,   3323376,   713948,  113816,   13012,   1006,    47,    1;
  9: 6625109, 59625981, 249215921, 642205132, 1142342460, 1488135236, 1470006144, 1124823234, 675266038, 320148710, 120059622, 35503348, 8212400, 1465092, 197086, 19287, 1293, 53, 1;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (16*n^2-42*n+31-(8*n-13)*x+x^2)*P(n-1,x) - (32*n-64+(32*n-78)*x+(4*n-19)*x^2-x^3)*d/dx P(n-1,x) + (16+32*x+18*x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 42*n + 31 - (8*n - 13)*x + x^2)*P[n - 1, x] - (32*n - 64 + (32*n - 78)*x + (4*n - 19)*x^2 - x^3)*D[P[n - 1, x], x] + (16 + 32*x + 18*x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}])/(2*n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^2 - 42*n + 31 - (8*n - 13)*x + x^2)*v[i - 1] - (32*n - 64 + (32*n - 78)*x + (4*n - 19)*x^2 - x^3)*deriv(v[i - 1], x) + (16 + 32*x + 18*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n - 1)); 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