Real-rooted atlas

A038337 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(46 + 22\,x + 23\,n + 11\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 506 + 242\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0506242748
11745716698399338148
2535348768108367356585641729376
3153912552944414021122970673486080525573498480
44247986381015822830971656620464705340111125190106293662998737984
51139876345332709495126391091789552493918658089455777951711327926136410197118949940032
62996246393361003091183864143921778554411472025826805486621047601574421691922509947624817148710484622054812672
777527875428192966283929426449653013601268474941869229522839326392133010863509674248259779579166835497830693621221529219176793596584704
81981267927609308528066297100901631456161184520182061049871316013060901403811806246518062692601991643440278804082250529764048809517203702593742460106678869204311040
950126078568515292397334192407253051594801097460445658020651677756405507346758607309031607381397224556125971447597634103442697946767640617440392626805656216842405303138428376721249789708241232896
101257708880446383646616642371044018281582240566988786980227017124828825958021714681505365961201453939544272329576695362390738940232237546158326967160568045161216660609055792425193140866206231974996376611405206529264927360220274688
11313379129377890590317985236990383286052473089929529647307027542013369313217938081158622126305279985621039891053466490323465222580515718822814205260267952636168424613758009453798590256064504424962012949783168026190211260681015674948448795257871103341258157768113451008
12776215997382159769824826038592419515090621384863248259513026004242852830491885617603629036751471855889061302499685561490202623538159400238703723065848876234896771345814101627345126015808824623472656131926062094410138994339336814441481812969643523666628144512073034495166531649767016537412495298699817077129216
131912817993548893718485128075639568056362020303981481838746099949761576167478654273216430220100176792577902817261485958212798571244339022906874135294098057214729537572913408734722664360157234474590900741584555013273102898284514290119882617678002455776520849150900522166222024928332598450416135183419169620705403062658722541234765455185881207622095421440
1446927801441732859226832336655966864605294453360112706562819889598577864023358026787310453038600803351369060788021522929680352622309874217916760427228107575424756500611338401728354016801797242560560826604094992163898615920307480750311094783543120882336066110098074515041918121882848039292424030578754654099177125603374912721020979415202305840989517439352073519567781715457616508074625129761327284224
Technical details
Node id
n-a8e1eaee11dec2de · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2f74b0bddfe64ff1

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:              506,               242;
  1:            17457,             16698,              3993;
  2:           535348,            768108,            367356,             58564;
  3:         15391255,          29444140,          21122970,           6734860,            805255;
  4:        424798638,        1015822830,         971656620,         464705340,         111125190,          10629366;
  5:      11398763453,       32709495126,       39109178955,       24939186580,        8945577795,        1711327926,         136410197;
  6:     299624639336,     1003091183864,     1439217785544,     1147202582680,      548662104760,      157442169192,       25099476248,       1714871048;
  7:    7752787542819,    29662839294264,    49653013601268,    47494186922952,    28393263921330,    10863509674248,     2597795791668,     354978306936,     21221529219;
  8:  198126792760930,   852806629710090,  1631456161184520,  1820610498713160,  1306090140381180,   624651806269260,   199164344027880,   40822505297640,   4880951720370,   259374246010;
  9: 5012607856851529, 23973341924072530, 51594801097460445, 65802065167775640, 55073467586073090, 31607381397224556, 12597144759763410, 3442697946767640, 617440392626805, 65621684240530, 3138428376721;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (23*n+46+(11*n+22)*x)*P(n-1,x) for n >= 1, with P(0,x) = 506 + 242*x.
%t Mathematica
P[0, x_] := 506 + 242*x;
P[n_, x_] := P[n, x] = Expand[Together[((23*n + 46 + (11*n + 22)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 506 + 242*x; for(n = 1, N, my(i = n + 1); v[i] = ((23*n + 46 + (11*n + 22)*x)*v[i - 1])/(n + 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.