Real-rooted atlas

A272867 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - x^{2} + 6\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(36 + 72\,x + 48\,x^{2} + 12\,x^{3} + x^{4} + 36\,n x + 48\,n x^{2} + 18\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1612321
236144144485377
321612962268172863010876253
412961036825920311042052077761680192998865
577767776025272041472039960024105693240230403510300111513733
6466565598722239488466560059292004945536280627210990082964605400063364321322648873
727993639191041861574447029248742996807936185659757264324311041280512836741607567561088641037458815333049821
8167961626873856147806208438939648829543680108167270410180961287101665283725714881479513604444070410015488166420819756815840768174831635809
910077696181398528113374080038698352648515653120131211601921484164684812684460032833803372842592262401698488352528519168127606752236234883285360331776229509721969337111285
10604661761209323520846526464032651735040820072512001458444165121928199168001953729331201547304249609699782400048492753408193995648006208319520158288256031881600050015232597618052488031920120021986218441689
113627970567981535232618568980482660511744007510402944001514496310272229264056806426888905359362495061169920185773927680011202571226885502018908162206567059847224541632019252015200414952243271602898497382736101950207920004296614522313923707676749
12217678233652242776064444063596544210712530124866047204044801489354474291225408773992448338736355246083606805886745631141289963520220321411706881286309055283262241927621122501156496384834550801920230676221952525683299689801399168147819513617734464016515576114998456304172825195345482950033
131306069401633957804441631410969108481629974613196856171868180480140075943321600265954065721344397331465895936477723816700416469643509969920381808543843584258758469844992146993230185984702317388810242826792888192095836470312962731795136928652282652736129697538256212904806402850159312305791200256398481617408721502028272725875959330277
147836416409621941965946882194196594688012360640816742446451446659072012693427300270082654896613031936439444174189363258902712366417926499350836858880597438309371904046152334998282243015117714196224167307772520448079070489653248031868147146752010952350671225632050232637696796249149696016716437337602945808946084316956300851935869445038225923841110022145769072023522937859388324031625
Technical details
Node id
n-f6ff745650619af7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-236414a5083672bc

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:        6,        12,          3;
  2:       36,       144,        144,         48,          5;
  3:      216,      1296,       2268,       1728,        630,         108,           7;
  4:     1296,     10368,      25920,      31104,      20520,        7776,        1680,         192,          9;
  5:     7776,     77760,     252720,     414720,     399600,      241056,       93240,       23040,       3510,        300,         11;
  6:    46656,    559872,    2239488,    4665600,    5929200,     4945536,     2806272,     1099008,     296460,      54000,       6336,       432,        13;
  7:   279936,   3919104,   18615744,   47029248,   74299680,    79361856,    59757264,    32431104,   12805128,    3674160,     756756,    108864,     10374,      588,      15;
  8:  1679616,  26873856,  147806208,  438939648,  829543680,  1081672704,  1018096128,   710166528,  372571488,  147951360,   44440704,  10015488,   1664208,   197568,   15840,    768,    17;
  9: 10077696, 181398528, 1133740800, 3869835264, 8515653120, 13121160192, 14841646848, 12684460032, 8338033728, 4259226240, 1698488352, 528519168, 127606752, 23623488, 3285360, 331776, 22950, 972, 19;
%F Formula
The row polynomials satisfy (6+6*n*x+(2*n-1)*x^2)*P(n,x) = (36+(36*n+72)*x+(48*n+48)*x^2+(18*n+12)*x^3+(2*n+1)*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[((36 + (36*n + 72)*x + (48*n + 48)*x^2 + (18*n + 12)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(6 + 6*n*x + (2*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] = ((36 + (36*n + 72)*x + (48*n + 48)*x^2 + (18*n + 12)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(6 + 6*n*x + (2*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

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.