Real-rooted atlas

A038329 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 75\,x + 36\,x^{2} + 45\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(375 + 2025\,x + 2295\,x^{2} + 729\,x^{3} + 675\,n x + 1215\,n x^{2} + 486\,n x^{3} + 135\,n^{2} x^{2} + 81\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 225 + 1080\,x + 729\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
022510807292034
1337524300328051166472144
2506254860009841506998401640252384640
37593759112500246037502624400012301875212576475147264
411390625164025000553584375787320000553584375191318760260406092287263744
5170859375287043750011625271875206671500001937545312510044234900273426394530611001667793780736
62562890625492075000002325054375004960116000005812635937504017693960001640558367003673320192034867844011967596240896
7384433593758303765625004484033437500111602610000001569411703125013559717115000738251265150024794911296004707158941353874204890056138410229760
857665039062513839609375000840756269531252391484500000003923529257812504067915134500002768442244312501239745564800003530369206012558113073350004219009125211579140457168896
986497558593752283535546875001541386494140625493243678125000092483189648437501118676661987500091358594062312505113950454800000194170306330687547943285513750069613650565965451887258369643890990473281536
101297463378906253736694531250000277449568945312509864873562500000020808717670898437528765971308250000027407578218693750018410221637280000087376637848809375287659713082500006265228550936850813397065065280477305916652891207453987104620544
11194619506835937560721286132812500491842417675781250192365034468750000045085554953613281257011705506385937500763496821806468750059833220321160000003407688876103565625140234110127718750040723985581089525079306213843864800930746537473135549820570235248432923093213605003264
122919292602539062598088231445312500086072423093261718753672423385312500000094679665402587890625163606461815671875000200417915724198046875179499660963480000000119269110663624796875588983262536418750002138009243007200062555514349690705360009772838643467922751046231974940216405147278302366225890731273965419888640
1343789389038085937515764180053710937500148971501507568359375688579384746093750000193662951959838867187536811453908526171875005010447893104951171875504842796459787500000038336499856165113281252208687234511570312500962104159353240028125312268217010217650000732962898260094206251177010971807743450011581376180324006255270812981623014423933390570135060742144
1465684083557128906252522268808593750000002553797168701171875000127122347953125000000003873259039196777343750080315899436784375000000120250749434518828125000134624745722610000000000115009499568495339843750757264194689681250000003848416637412960112500014988874416490447200000439777738956056523750094160877744619476000013897651416388807500012649951155895234560535522834578182049639149320222354370985984
Technical details
Node id
n-8121b088b939d544 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b34740d661cfea25

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:           225,            1080,              729;
  1:          3375,           24300,            32805,            11664;
  2:         50625,          486000,           984150,           699840,           164025;
  3:        759375,         9112500,         24603750,         26244000,         12301875,           2125764;
  4:      11390625,       164025000,        553584375,        787320000,        553584375,         191318760,         26040609;
  5:     170859375,      2870437500,      11625271875,      20667150000,      19375453125,       10044234900,       2734263945,        306110016;
  6:    2562890625,     49207500000,     232505437500,     496011600000,     581263593750,      401769396000,     164055836700,      36733201920,       3486784401;
  7:   38443359375,    830376562500,    4484033437500,   11160261000000,   15694117031250,    13559717115000,    7382512651500,    2479491129600,     470715894135,     38742048900;
  8:  576650390625,  13839609375000,   84075626953125,  239148450000000,  392352925781250,   406791513450000,  276844224431250,  123974556480000,   35303692060125,   5811307335000,   421900912521;
  9: 8649755859375, 228353554687500, 1541386494140625, 4932436781250000, 9248318964843750, 11186766619875000, 9135859406231250, 5113950454800000, 1941703063306875, 479432855137500, 69613650565965, 4518872583696;
%F Formula
The row polynomials satisfy (25+(45*n+75)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (375+(675*n+2025)*x+(135*n^2+1215*n+2295)*x^2+(81*n^2+486*n+729)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 1080*x + 729*x^2.
%t Mathematica
P[0, x_] := 225 + 1080*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((375 + (675*n + 2025)*x + (135*n^2 + 1215*n + 2295)*x^2 + (81*n^2 + 486*n + 729)*x^3)*P[n - 1, x])/(25 + (45*n + 75)*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] = 225 + 1080*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((375 + (675*n + 2025)*x + (135*n^2 + 1215*n + 2295)*x^2 + (81*n^2 + 486*n + 729)*x^3)*v[i - 1])/(25 + (45*n + 75)*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.