Real-rooted atlas

A038337 → 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

\(P_{n}(x) = \left(45 + 22\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2025 + 1980\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0202519804844489
1911251336506534010648300763
2410062580190005880600191664023425620151121
31845281254510687504410450002156220005270760051536321350125107
483037656252435771250029770537500194059800007115526000139148064011337990490458382169
53736694531251278779906250187554386250015282209250007471302300002191582008003571466976024943578886060711605323
616815125390625657658237500001125326317500001100319066000006724172070000026298984096000642864055680089796883968054875873536406067677556641
77566806425781253329394827343750651081655125000074271536955000005446579376700000266277213972000086786647516800018183869003520022224728782080120726921779227206534396294947
83405062891601562516646974136718750036623343100781250047745988042500000040849345325250000023964949257480000097634978456400000272758035052800005000563975968000543271148006400265599227914241822837804551761449
915322783012207031258240252197675781250201428387054296875002954283010129687500028886322765712500000197710831374210000009665862867183600000337538068377840000082509305603472000013445960913158400013147161781754880584318301411328122130132904968017083
10689525235549316406254045214715222656250001087713290093203125000177256980607781250000019498267866855937500001525197842029620000000869927658046524000000364541113848067200000111387562564687200000242027296436851200003549733681073817600315531882762117120128550026310492168182718904632857144561
11310286355997192382812519720421736710449218750578465704276839843750001036953336555520312500001267387411345635937500001115300921984159625000007270109713674522000000035542758600186552000000130323448200684024000003539649210388948800000692198067809394432000922930757079192576007520176539163791360282810057883082752548242166610401428685587
121396288601987365722656259556819764713525390625003036944947453409179687500593891456390888906250000079845406914775064062500007807106453889117375000000572521139951868607500000031988482740167896800000001368396206107182252000000445995800509007548800000109021195679979623040000193815458986630440960002368855609836594278400178170336466342133760622182127342782054436732225162896895721934329
13628329870894314575195312546077523865583068847656250157687526117773168945312500334063944219875009765625000489960451522483347656250000526979685637515422812500000429390854963901455625000000269902823120166629250000000131952491303192574300000000501745275572633492400000001471786141679724911040000032706358703993886912000005329925122132337126400006013248855739047014400041997293595637788672001368800680154120519682461059085914092013369600043
14282748441902441558837890625221172114554798730468750000081096442003426201171875000001850200306448538515625000000029397627091349000859375000000344932157871828276750000000003091614155740090480500000000021592225849613330340000000000118757242172873316870000000005160808548747087350400000000176614337001566989324800000047097156533751197153280000095938652198382068275200000144317972537737128345600001511902569442960392192000985536489710966774169603011361496339065143296164890958756244164895763202881
Technical details
Node id
n-9ed77544d1e5ce22 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a7a31a2ddce7ca1a

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:                2025,                1980,                  484;
  1:               91125,              133650,                65340,                10648;
  2:             4100625,             8019000,              5880600,              1916640,               234256;
  3:           184528125,           451068750,            441045000,            215622000,             52707600,              5153632;
  4:          8303765625,         24357712500,          29770537500,          19405980000,           7115526000,           1391480640,           113379904;
  5:        373669453125,       1278779906250,        1875543862500,        1528220925000,         747130230000,         219158200800,         35714669760,          2494357888;
  6:      16815125390625,      65765823750000,      112532631750000,      110031906600000,       67241720700000,       26298984096000,       6428640556800,        897968839680,        54875873536;
  7:     756680642578125,    3329394827343750,     6510816551250000,     7427153695500000,     5446579376700000,     2662772139720000,     867866475168000,     181838690035200,     22224728782080,      1207269217792;
  8:   34050628916015625,  166469741367187500,   366233431007812500,   477459880425000000,   408493453252500000,   239649492574800000,   97634978456400000,   27275803505280000,   5000563975968000,    543271148006400,    26559922791424;
  9: 1532278301220703125, 8240252197675781250, 20142838705429687500, 29542830101296875000, 28886322765712500000, 19771083137421000000, 9665862867183600000, 3375380683778400000, 825093056034720000, 134459609131584000, 13147161781754880, 584318301411328;
%F Formula
The row polynomials satisfy P(n,x) = (45+22*x)*P(n-1,x) for n >= 1, with P(0,x) = 2025 + 1980*x + 484*x^2.
%t Mathematica
P[0, x_] := 2025 + 1980*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[(45 + 22*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] = 2025 + 1980*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = (45 + 22*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

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.