Real-rooted atlas

A038319 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(26 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 676 + 520\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
06765201001296
117576202807800100046656
2456976703040405600104000100001679616
31188137622848800175760006760000130000010000060466176
43089157767128825606854640003515200001014000001560000010000002176782336
58031810176216241043202495088960015994160000615160000014196000001820000001000000078364164096
6208827064576642544814080864964172800665357056000319883200000984256000001892800000020800000001000000002821109907456
754295036789761879443581184028914516633600259489251840001497053376000057578976000001476384000000243360000000234000000001000000000101559956668416
814116709565337654295036789760093972179059200096381722112000064872312960000029941067520000095964960000000210912000000003042000000000260000000000100000000003656158440062976
93670344486987776155283805218713602986227023436800034456465655040000265049735808000001427190885120000054891957120000001508020800000000290004000000000371800000000002860000000000100000000000131621703842267136
109542895666168217644044133843853312093170283131228160011944908093747200001033693969651200000636119365939200000285438177024000000941004979200000002262031200000000038667200000000004461600000000003120000000000010000000000004738381338321616896
1124811528732037365761240576436601868288028628686998504652800403737893568655360003882095130467840000026876043210931200000137825862620160000005300994716160000000152913309120000000032673784000000000050267360000000000527280000000000033800000000000010000000000000170581728179578208256
1264509974703297150976347361402248523120640868403505621307801600133600539326355046400014130826274902937600001086986636530995200000627107674921728000000275651725240320000000927674075328000000002378651475200000000045743297600000000006397664000000000006151600000000000036400000000000001000000000000006140942214464815497216
131677259342285725925376967649620549457264640026052105168639234048000434201752810653900800005010020224738314240000042392478824708812800000271746659132748800000001343802160546560000000051684698482560000000001546123458880000000000356797721280000000000623772240000000000007997080000000000000709800000000000000390000000000000001000000000000000221073919720733357899776
1443608742899428874059776268361494765716148060160774119696439565811712000138944560899409248256000017368070112426156032000001603206471916260556800000113046610199223500800000062113522087485440000000026876043210931200000000091883908413440000000000247379753420800000000005189785036800000000000831696320000000000000984256000000000000008112000000000000000416000000000000000100000000000000007958661109946400884391936
Technical details
Node id
n-a499436c313e0808 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-3dee56618ab9ea64

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              676,               520,               100;
  1:            17576,             20280,              7800,              1000;
  2:           456976,            703040,            405600,            104000,             10000;
  3:         11881376,          22848800,          17576000,           6760000,           1300000,            100000;
  4:        308915776,         712882560,         685464000,         351520000,         101400000,          15600000,          1000000;
  5:       8031810176,       21624104320,       24950889600,       15994160000,        6151600000,        1419600000,        182000000,         10000000;
  6:     208827064576,      642544814080,      864964172800,      665357056000,      319883200000,       98425600000,      18928000000,       2080000000,       100000000;
  7:    5429503678976,    18794435811840,    28914516633600,    25948925184000,    14970533760000,     5757897600000,    1476384000000,     243360000000,     23400000000,     1000000000;
  8:  141167095653376,   542950367897600,   939721790592000,   963817221120000,   648723129600000,   299410675200000,   95964960000000,   21091200000000,   3042000000000,   260000000000,   10000000000;
  9: 3670344486987776, 15528380521871360, 29862270234368000, 34456465655040000, 26504973580800000, 14271908851200000, 5489195712000000, 1508020800000000, 290004000000000, 37180000000000, 2860000000000, 100000000000;
%F Formula
The row polynomials satisfy P(n,x) = (26+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 520*x + 100*x^2.
%t Mathematica
P[0, x_] := 676 + 520*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(26 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 676 + 520*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 10*x)*v[i - 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.