Real-rooted atlas

A038226 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 440\,x + 256\,x^{2} + 264\,n x + 256\,n x^{2} + 64\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 8712\,x + 11968\,x^{2} + 4608\,x^{3} + 2904\,n x + 6336\,n x^{2} + 3072\,n x^{3} + 704\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 704\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01217045761401
113311161619008819240147
2146411703684181763604481024001066033
31610512342560766656099123205632000117964826894139
41771561309217921264982402180710401858560007785676812845056653820457
519487171396829664194807289641978675204770304000299748556898906931213421772815453333859
6214358881498871577628571735808738824683521049466880008792624332843519049728118111600641358954496357219374433
72357947691617353577284040859778561219060727808207794442240021761745223681436128641024584652423168134536495104134217728008110098287947
8259374246017545432611205556182195520191566685798403809564774400047875839492096394935376281602143725551616073995072307201476395008000129922760704181401436840921
9285311670611912997345955274699782850880289744612270080658510482432000965496096423936955743610601472648476979363840298446791639040893218979840001572065404518412369505812484006813143321843
1031384283767211095596815146249860371336316164249587646627840108654229601280001820649781827993621026359433232384171197922552053769848744124088320393016351129600010375631669821441632774767247361161359156838487557757227656081
113452271214393113055862047159361281848273721100860769103346778112172639498144256000325441148501753856429538485564604416408021715415728128281674081948925952140503345528832000494571776261488641167433958581862416607435942789121077521395220481895645983050717787
123797498335832411546617504048108816450386179420793685076744685489356826586482714215424005568659652141121536826861584711863500889764777391460188167229634770022432768432750304228802560019041013386067312645992827654053560321278772567594762241659382948639539298956046499840040709501306383899273
13417724816941565118227992011995568020879336304649468801169805239425478656039879724071323136000918828842603285053441515912905305083084801851398533698866380801704128195791002009601190063336629207040006283534417402213171224720414072970936320703324912177119232013689909326276198401632774767247360009007199254740992868015631644447954819
1445949729863572161213875106274081331226248308497273617920158373632414526341120584902619712739328000147012614816525608550426680067133369462292483620512688122227589120374908203074020442112029921592463820062720001843170095771315863552870158575368576958464309462961357932462080803141347141536972801436841795177676800015852670688344145928134626826937958418390736886664643813057
Technical details
Node id
n-f95ad06430ea3396 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-46efc6967b68e78c

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:          121,           704,            576;
  1:         1331,         11616,          19008,            8192;
  2:        14641,        170368,         418176,          360448,          102400;
  3:       161051,       2342560,        7666560,         9912320,         5632000,         1179648;
  4:      1771561,      30921792,      126498240,       218071040,       185856000,        77856768,        12845056;
  5:     19487171,     396829664,     1948072896,      4197867520,      4770304000,      2997485568,       989069312,       134217728;
  6:    214358881,    4988715776,    28571735808,     73882468352,    104946688000,     87926243328,     43519049728,     11811160064,      1358954496;
  7:   2357947691,   61735357728,   404085977856,   1219060727808,   2077944422400,   2176174522368,   1436128641024,    584652423168,    134536495104,    13421772800;
  8:  25937424601,  754543261120,  5556182195520,  19156668579840,  38095647744000,  47875839492096,  39493537628160,  21437255516160,   7399507230720,  1476395008000,   129922760704;
  9: 285311670611, 9129973459552, 74699782850880, 289744612270080, 658510482432000, 965496096423936, 955743610601472, 648476979363840, 298446791639040, 89321897984000, 15720654045184, 1236950581248;
%F Formula
The row polynomials satisfy (121+(264*n+440)*x+(64*n^2+256*n+256)*x^2)*P(n,x) = (1331+(2904*n+8712)*x+(704*n^2+6336*n+11968)*x^2+(512*n^2+3072*n+4608)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 704*x + 576*x^2.
%t Mathematica
P[0, x_] := 121 + 704*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1331 + (2904*n + 8712)*x + (704*n^2 + 6336*n + 11968)*x^2 + (512*n^2 + 3072*n + 4608)*x^3)*P[n - 1, x])/(121 + (264*n + 440)*x + (64*n^2 + 256*n + 256)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 704*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (2904*n + 8712)*x + (704*n^2 + 6336*n + 11968)*x^2 + (512*n^2 + 3072*n + 4608)*x^3)*v[i - 1])/(121 + (264*n + 440)*x + (64*n^2 + 256*n + 256)*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.