Real-rooted atlas

A038305 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 13\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 312\,x + 169\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144312169625
1172856166084219715625
2207368985614601610545628561390625
324883213478402920320316368017136603712939765625
4298598419408896525657607592832061691760267330964826809244140625
535831808271724544883104768159449472017273692801122790032405451956627485176103515625
6429981696372650803214129676288306142986244145686272035929281024194616938886023857632815730721152587890625
751597803525030785843221800071987255105737523289546823475297009058764870062097996832528831212888098917868106044993733814697265625
86191736422467077144576032700107980809446697861120179093646950402328217410355221018629399040130115324851205285935072080127253992476013785849184995367431640625
9743008370688885418308403247960158371840155870514708480337719448535040512207830278144554891816134656429380572008960232581143171520839876350341601819732092406817921603940372384185791015625
108916100448256115909305827328690626280554496249392823533568060789500736307201053684679429324813317403587231744123661604738580488372921154174720403140648163968013102071065328962580710967413282329808512248159604644775390625
111069932053790721506820975755264979433634240921638905280471236608105368467942932480205468512488718336296787851372593152321520172320309248261235140010251264157224852783947520681307695397105922012954554582358436345012791070363028751065922531490116119384765625
121283918464548864194727633789911041371207087937290245941897381061591041770190261441265664383541223312274227262325448788244561927716484135687421952731458392028703539252827550535406366722861492320667844864112725455056612070430529810744499102450883017907498504393737638569928937252902984619140625
13154070215745863682503641005870284801898594429451632640891284607159238656028966749732675256320690374201962093608961246508975764891238401736208930529669939201880893008073809100801584826516062191001601030137235440424151045072645477547543168018317886446699461440457947161167486536070872774942587202051185893014090757931322574615478515625
14184884258895036416320466048751396454426037866461050961920131635880441979863040463467995722804101120120501678887929066291223932972334685911777283703912385129962536960451414321937714184192043469527297705810329603296439153409357283328194789586337825657651287925854944157414912029308618314719138304068037863944883713920982769145870542534466541660918317984123283064365386962890625
Technical details
Node id
n-c39ad02ee0de214a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-42b2d489dcb16cdd

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          144,           312,            169;
  1:         1728,          5616,           6084,            2197;
  2:        20736,         89856,         146016,          105456,           28561;
  3:       248832,       1347840,        2920320,         3163680,         1713660,          371293;
  4:      2985984,      19408896,       52565760,        75928320,        61691760,        26733096,         4826809;
  5:     35831808,     271724544,      883104768,      1594494720,      1727369280,      1122790032,       405451956,        62748517;
  6:    429981696,    3726508032,    14129676288,     30614298624,     41456862720,     35929281024,     19461693888,      6023857632,       815730721;
  7:   5159780352,   50307858432,   218000719872,    551057375232,    895468234752,    970090587648,    700620979968,    325288312128,     88098917868,    10604499373;
  8:  61917364224,  670771445760,  3270010798080,   9446697861120,  17909364695040,  23282174103552,  21018629399040,  13011532485120,   5285935072080,  1272539924760,   137858491849;
  9: 743008370688, 8854183084032, 47960158371840, 155870514708480, 337719448535040, 512207830278144, 554891816134656, 429380572008960, 232581143171520, 83987635034160, 18197320924068, 1792160394037;
%F Formula
The row polynomials satisfy P(n,x) = (12+13*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 312*x + 169*x^2.
%t Mathematica
P[0, x_] := 144 + 312*x + 169*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 13*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 312*x + 169*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 13*x)*v[i - 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

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.