Real-rooted atlas

A038330 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 15\,x + 4\,x^{2} + 9\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(108 + 324\,x + 204\,x^{2} + 36\,x^{3} + 108\,n x + 108\,n x^{2} + 24\,n x^{3} + 12\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 384\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144384144672
1172869125184102414848
220736110592124416491526400311296
3248832165888024883201474560384000368646291456
42985984238878724478976035389440138240002654208200704123731968
5358318083344302087524679687431782403870720001114767361685913610485762382364672
64299816964586471424120394874881426902220892897280003567255552809238528100663296530841645097156608
75159780352619173642241857520926722568423997442006581248009631589990429132587008543581798457330892826214400841813590016
861917364224825564856320278628139008044030125670404013162496000231158159769687397761024021743271936034398535680314572800012687769615530601742336
97430083706881089745610342440865460387840726497073561607567677849600050854795149312230730089103367175279738880151353556992020761804800016747855872603979776283673999966208
10891610044825614265760717209658846262958489611623953176985601362182012928000104615578592870455375221384806420664805647974454487280517120996566630400012058456227848697308774428353495045136918324969472
111069932053790721854548893237248834547001956761618133366956097536236111548907520002040003782560972812340763622899712537284946847334417000031521341443886609858560006270397238476867839008440324423145226241315333734492323792361095168
1212839184645488642396647800491212811683658027394662427694596805676236839666740216463360038080070607804825625915603608089395212894838724336025647600088259756032130590091247616002633566840160256379898447265792371544199004162209760673792603979776001648317463617601536
13154070215745863683081404314917273601617737265331568640415418952085143552064909211263303680006854412709404868608518312072161787904029013387129756057601224002269536583680391770273742848000948084062457692161709543012696064022292651940249601988784606412801087163596800027487790694429255383179398742016
1418488425889503641639441975230941102082218611106740436992061354183692575047680103854738021285888000119640658200521342976995159178550632775686189522587681292288029376054468878008320107456989369466880003033868999864614912656464516875288576107004729313198080127282214810419201043677052928000527765581332481241245548544516508834063867445248
Technical details
Node id
n-70ef400185a6f340 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1e916cb7d7a3299c

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:          144,            384,            144;
  1:         1728,           6912,           5184,           1024;
  2:        20736,         110592,         124416,          49152,           6400;
  3:       248832,        1658880,        2488320,        1474560,         384000,          36864;
  4:      2985984,       23887872,       44789760,       35389440,       13824000,        2654208,         200704;
  5:     35831808,      334430208,      752467968,      743178240,      387072000,      111476736,       16859136,       1048576;
  6:    429981696,     4586471424,    12039487488,    14269022208,     9289728000,     3567255552,      809238528,     100663296,       5308416;
  7:   5159780352,    61917364224,   185752092672,   256842399744,   200658124800,    96315899904,    29132587008,    5435817984,     573308928,     26214400;
  8:  61917364224,   825564856320,  2786281390080,  4403012567040,  4013162496000,  2311581597696,   873977610240,  217432719360,   34398535680,   3145728000,   126877696;
  9: 743008370688, 10897456103424, 40865460387840, 72649707356160, 75676778496000, 50854795149312, 23073008910336, 7175279738880, 1513535569920, 207618048000, 16747855872, 603979776;
%F Formula
The row polynomials satisfy (9+(9*n+15)*x+(n^2+4*n+4)*x^2)*P(n,x) = (108+(108*n+324)*x+(12*n^2+108*n+204)*x^2+(4*n^2+24*n+36)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 384*x + 144*x^2.
%t Mathematica
P[0, x_] := 144 + 384*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((108 + (108*n + 324)*x + (12*n^2 + 108*n + 204)*x^2 + (4*n^2 + 24*n + 36)*x^3)*P[n - 1, x])/(9 + (9*n + 15)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 384*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((108 + (108*n + 324)*x + (12*n^2 + 108*n + 204)*x^2 + (4*n^2 + 24*n + 36)*x^3)*v[i - 1])/(9 + (9*n + 15)*x + (n^2 + 4*n + 4)*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.