Real-rooted atlas

A139548 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + x^{2} + 6\,n x - 4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 32\,x + 48\,x^{2} + 32\,x^{3} + 8\,x^{4} + 48\,n x + 128\,n x^{2} + 112\,n x^{3} + 32\,n x^{4} + 32\,n^{2} x^{2} + 64\,n^{2} x^{3} + 32\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
186472144
264102434564096160010240
3512122886912016384019200011059225088573440
4409613107210321923670016716800082575365619712209715233177628311552
532768131072013271040629145601720320002972712963371827202516582401194393603276800039649281291845632
62621441258291215571353692274688032440320007474249728118688317441328755507210510663680576716800020934819844529848324430233655834574848
72097152117440512171756748812213813248524812288001511459389443085896253444606352424965101175439364198498304002540091473921099243192323225210060857545850884718592002319282339840

All 15 rows

Technical details
Node id
n-eac81999dec1bbfd · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-561a7d876fc231a6

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:         8,         64,           72;
  2:        64,       1024,         3456,          4096,           1600;
  3:       512,      12288,        69120,        163840,         192000,         110592,           25088;
  4:      4096,     131072,      1032192,       3670016,        7168000,        8257536,         5619712,         2097152,          331776;
  5:     32768,    1310720,     13271040,      62914560,      172032000,      297271296,       337182720,       251658240,       119439360,        32768000,         3964928;
  6:    262144,   12582912,    155713536,     922746880,     3244032000,     7474249728,     11868831744,     13287555072,     10510663680,      5767168000,      2093481984,       452984832,        44302336;
  7:   2097152,  117440512,   1717567488,   12213813248,    52481228800,   151145938944,    308589625344,    460635242496,    510117543936,    419849830400,    254009147392,    109924319232,     32252100608,      5754585088,      471859200;
  8:  16777216, 1073741824,  18119393280,  150323855360,   763363328000,  2638183661568,   6583245340672,  12283606466560,  17489744363520,  19193135104000,  16256585433088,  10552734646272,   5160336097280,   1841467228160,   452984832000,    68719476736,    4848615424;
  9: 134217728, 9663676416, 184817811456, 1752346656768, 10267656192000, 41399189766144, 122089277227008, 273366078455808, 475721046687744, 652566593536000, 710645020360704, 615073676525568, 421083425538048, 225395588726784, 92408905728000, 28037546508288, 5934705278976, 782757789696, 48452599808;
%F Formula
The row polynomials satisfy (1+(6*n-4)*x+(4*n^2-4*n+1)*x^2)*P(n,x) = (8+(48*n+32)*x+(32*n^2+128*n+48)*x^2+(64*n^2+112*n+32)*x^3+(32*n^2+32*n+8)*x^4)*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[((8 + (48*n + 32)*x + (32*n^2 + 128*n + 48)*x^2 + (64*n^2 + 112*n + 32)*x^3 + (32*n^2 + 32*n + 8)*x^4)*P[n - 1, x])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*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; for(n = 1, N, my(i = n + 1); v[i] = ((8 + (48*n + 32)*x + (32*n^2 + 128*n + 48)*x^2 + (64*n^2 + 112*n + 32)*x^3 + (32*n^2 + 32*n + 8)*x^4)*v[i - 1])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*x^2)); 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

Same fingerprint ?

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.