Real-rooted atlas

A038330 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 + 60\,x + 36\,x^{2} + 36\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(256 + 1728\,x + 2448\,x^{2} + 972\,x^{3} + 576\,n x + 1296\,n x^{2} + 648\,n x^{3} + 144\,n^{2} x^{2} + 108\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 1536\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0256153612963088
14096368646220827648130816
265536786432199065617694725184005130496
31048576157286405308416070778880414720008957952191070208
4167772163019898881274019840226492416019906560008599633921463132166854643712
5268435456563714457628538044416634178764807431782400048157949952163870801922293235712239017590784
642949672961030792151046088116142081623497637888237817036800020547391979521048773132288293534171136348285173768149728821248
76871947673618554258718721252412463513638963943309312684913065984007397061112627250341110349824211344603217925015306502144515978035200272880986226688
810995116277763298534883328025048249270272089060441849856018264348426240002367059556040704201364441399296011271712171622404012245201715208255648563200074920010711049000754808356864
917592186044416580542139465728489832430174208019593297206968320459217903288320006943374697719398470880283372552192495955335551385602353850518339584072649707356160001318592188514304106993205379072293150171380842496
102814749767106561013309916158361694047826593447936417990340415324160110212296789196800019044684885173207042268169067921670144190446848851732070411298482488030003204649581270794240001265848500973731842054269543278182415068209757552649445122496383680512
114503599627370496175640385467449344177835890285792460886941990806387425282547128636905881600049516180701450338304673970237325296271366602157426860045107247001687150204813312241778226081300480008776549606751207424213644032500930969631341876295709491220970668254298112301485656161537490944
1272057594037927936302641894959297331233196032853347926016177045508551188938752570556814666917478400123240271968054175334418871166645108295598082112690376595214434304175472965360764636364810831664528442261504004914867779780676157441595208776006951239683510290145119463014446974296889627770882888816545234944009545099568226987147264
1311529215046068469765188146770730811392061284983729257709568035409101710237787750401244851232000547225600029577665272333002080256503231110536221215948806338071129785643302912060162159552262161039360433266581137690460160002359136534294724555571295712526560417074380802808232116095570411520563691562675533250560693315970856386560003944197523094110208300033429479745296269312
14184467440737095516168854437155380584775681120639702477855260672069728692598622105108480265568262826783408128000688352937247022593867776128827164297272631282892818028291213612496506060801925189105672389153259520158451778244641082572800010065649212990824770437124900481359893354208296961797268554301165063372804810168001497883738112088744444269617479680001009714565912092213248534315508206655242249370947015886802117459968
Technical details
Node id
n-6b13d799f6d8c590 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e997b32cda4979c2

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            256,            1536,             1296;
  1:           4096,           36864,            62208,             27648;
  2:          65536,          786432,          1990656,           1769472,            518400;
  3:        1048576,        15728640,         53084160,          70778880,          41472000,           8957952;
  4:       16777216,       301989888,       1274019840,        2264924160,        1990656000,         859963392,         146313216;
  5:      268435456,      5637144576,      28538044416,       63417876480,       74317824000,       48157949952,       16387080192,        2293235712;
  6:     4294967296,    103079215104,     608811614208,     1623497637888,     2378170368000,     2054739197952,     1048773132288,      293534171136,       34828517376;
  7:    68719476736,   1855425871872,   12524124635136,    38963943309312,    68491306598400,    73970611126272,    50341110349824,    21134460321792,     5015306502144,     515978035200;
  8:  1099511627776,  32985348833280,  250482492702720,   890604418498560,  1826434842624000,  2367059556040704,  2013644413992960,  1127171217162240,   401224520171520,   82556485632000,    7492001071104;
  9: 17592186044416, 580542139465728, 4898324301742080, 19593297206968320, 45921790328832000, 69433746977193984, 70880283372552192, 49595533555138560, 23538505183395840, 7264970735616000, 1318592188514304, 106993205379072;
%F Formula
The row polynomials satisfy (16+(36*n+60)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (256+(576*n+1728)*x+(144*n^2+1296*n+2448)*x^2+(108*n^2+648*n+972)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 1536*x + 1296*x^2.
%t Mathematica
P[0, x_] := 256 + 1536*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((256 + (576*n + 1728)*x + (144*n^2 + 1296*n + 2448)*x^2 + (108*n^2 + 648*n + 972)*x^3)*P[n - 1, x])/(16 + (36*n + 60)*x + (9*n^2 + 36*n + 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 1536*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((256 + (576*n + 1728)*x + (144*n^2 + 1296*n + 2448)*x^2 + (108*n^2 + 648*n + 972)*x^3)*v[i - 1])/(16 + (36*n + 60)*x + (9*n^2 + 36*n + 36)*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. [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.