Real-rooted atlas

A038337 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(46 + 44\,x + 23\,n + 22\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1012 + 968\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
010129681980
1349146679231944133650
21070696307243229388489370248019000
33078251011777656016898376010775776025768160451068750
4849597276406329132077732529607435285440355600608068027942424357712500
522797526906130837980504312873431640399026985280286258489440109524987264174605052161278779906250
6599249278672401236473545611513742284352183552413228801755718735232010076298828288321273295974443900698828865765823750000
71550557508563811865135717705639722410881014475990699076723290858444548256069526461915187233251786133350490874446575616108654229601283329394827343750
839625358552186034112265188403601305164928947616029129767979410560417948844921977603997771560123264025493036035568640104505613561958402499047280829440265599227914240166469741367187500
9100252157137030589589336769629012041275840877968356010528330426844102401762350962754338880202287240942237158416124345292497164808813306743725158403161294810249241606719660466230272064275013155246088240252197675781250
1025154177608927672826466569484176073121265792453591029584036322739972612153280694869808171710758409305213083342909286489006386014584349696608118165317035929602908391225429302272092731314433977753601773990363084791808154260031572590592404521471522265625000
1162675825875578118067194094796153314420837847194362371784561612067221390901148700802597075908041768959520397465530274218553804844354849030601200931843636546628595874859008217402244100840344832092422210052531161088026521155928117637529646123749440204587008367653075248007577619720421736710449218750
12155243199476431953964193041543696780603624811078905986076104208032388564528787016988165769291760470993884499616015997987593537296790643220403230554076552428646419516133573464528410009614000704520094118207180874399879092287584675840284660406961795976151047425923659872938508288118384290229858439987287105497827989487616955681976471352539062500
133825635987097787436970512302558272225448081203185185470996879959809201218679658468371462883520320565736249289015236752061325619108559637697465608798893176445513234853760961866583263608900207616080504050990541179691289605133591657367843342632960245519601004549029430272085398122088538792845312020421290064650580897792030051396750656373227520205320102023118077952046077523865583068847656250
1493855602883465718453664134662386745842117781344090165250255911678862291203737284285969672486176128010724380994521668873374976022567827831949946672667340835977696543688320782513152044245862830126009409550336042322129663598791609135104031486028831856105834815488018070242633934808566068019278566272321455689417687040250501158126380459012915205529457002120772673863680755577975445074526863360481817839414250422927362211721145547987304687500000
Technical details
Node id
n-eea50a67a563377c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-efa2c9a7f1c36a29

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:              1012,               968;
  1:             34914,             66792,              31944;
  2:           1070696,           3072432,            2938848,              937024;
  3:          30782510,         117776560,          168983760,           107757760,            25768160;
  4:         849597276,        4063291320,         7773252960,          7435285440,          3556006080,           680279424;
  5:       22797526906,      130837980504,       312873431640,        399026985280,        286258489440,        109524987264,         17460505216;
  6:      599249278672,     4012364735456,     11513742284352,      18355241322880,      17557187352320,      10076298828288,       3212732959744,       439006988288;
  7:    15505575085638,   118651357177056,    397224108810144,     759906990767232,     908584445482560,     695264619151872,     332517861333504,     90874446575616,     10865422960128;
  8:   396253585521860,  3411226518840360,  13051649289476160,   29129767979410560,   41794884492197760,   39977715601232640,   25493036035568640,  10450561356195840,   2499047280829440,   265599227914240;
  9: 10025215713703058, 95893367696290120, 412758408779683560, 1052833042684410240, 1762350962754338880, 2022872409422371584, 1612434529249716480, 881330674372515840, 316129481024924160, 67196604662302720, 6427501315524608;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (23*n+46+(22*n+44)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1012 + 968*x.
%t Mathematica
P[0, x_] := 1012 + 968*x;
P[n_, x_] := P[n, x] = Expand[Together[((23*n + 46 + (22*n + 44)*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] = 1012 + 968*x; for(n = 1, N, my(i = n + 1); v[i] = ((23*n + 46 + (22*n + 44)*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.