A089503 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(-1 + 2\,x^{2} + n - 2\,n x - 3\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3} - n - 3\,n x - 3\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	4														9
3	43	72	30													145
4	529	1368	1176	336												3409
5	8501	29920	39420	23040	5040											105921
6	169021	755100	1347720	1201200	534600	95040										4102681
7	4010455	21758184	49142730	59142720	39999960	14414400	2162160									190630609
8	110676833	707280112	1935818640	2941528800	2679986400	1463978880	443963520	57657600								10340890785
9	3484717129	25651375488	82568248056	151793703936	174319387200	128051159040	58757287680	15397724160	1764322560							641787925249
10	123320412181	1027994161140	3807078240240	8221185413280	11408087310480	10549156020480	6500479910400	2573936870400	594255916800	60949324800						44866443580201
11	4847038223171	45146191347400	189164857009590	469543542652800	764606705108400	853490307922560	661377093300000	351311908147200	122420235168000	25270527744000	2346549004800					3489524955627921
12	209536628422705	2157204484374984	10092243524765160	28321840204709040	52972243427869920	69335554162089600	64806191066010240	43254178224364800	20202955324675200	6289060767897600	1174305952512000	99638080819200				298914951848510449
13	9882471447634813	111457337689612512	576023870271606900	1803825132485376000	3812036220693558000	5727441691215129600	6273262628006774400	5047004592774144000	2960059295072880000	1234249282506240000	347300985455424000	59213488029696000	4626053752320000			27966383049400396225
14	505033804901100749	6193383971188980748	35048218917373769016	121184111481654036816	285637147007891679000	484658635534566292800	609035036983294521600	573888760488871027200	405500405068097539200	212195884348768320000	79933705240701312000	20526045407771904000	3220427319677568000	233153109116928000		2837759948683874979129
15	27802319803528367791	368377504919100489960	2265817949479253543010	8575064733626068816320	22307744750841588958920	42199148812048954165440	59861105031666620682000	64687070089261331712000	53510205962759770224000	33719877037378756224000	15934020292612854432000	5475074428677934080000	1293166861832926080000	187928263392629760000	12677700308232960000	310425081738609550495441

Technical details

Node id: n-f573c9322e5df2c8 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-11ac09149027f91e

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:            1;
   2:            5,             4;
   3:           43,            72,            30;
   4:          529,          1368,          1176,           336;
   5:         8501,         29920,         39420,         23040,           5040;
   6:       169021,        755100,       1347720,       1201200,         534600,          95040;
   7:      4010455,      21758184,      49142730,      59142720,       39999960,       14414400,       2162160;
   8:    110676833,     707280112,    1935818640,    2941528800,     2679986400,     1463978880,     443963520,      57657600;
   9:   3484717129,   25651375488,   82568248056,  151793703936,   174319387200,   128051159040,   58757287680,   15397724160,   1764322560;
  10: 123320412181, 1027994161140, 3807078240240, 8221185413280, 11408087310480, 10549156020480, 6500479910400, 2573936870400, 594255916800, 60949324800;

%F Formula

The row polynomials satisfy P(n,x) = (n^2+n-1+(2*n^2-2*n)*x+(n^2-3*n+2)*x^2)*P(n-1,x) - (n+(3*n-1)*x+(3*n-2)*x^2+(n-1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^2 + n - 1 + (2*n^2 - 2*n)*x + (n^2 - 3*n + 2)*x^2)*P[n - 1, x] - (n + (3*n - 1)*x + (3*n - 2)*x^2 + (n - 1)*x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n^2 + n - 1 + (2*n^2 - 2*n)*x + (n^2 - 3*n + 2)*x^2)*v[i - 1] - (n + (3*n - 1)*x + (3*n - 2)*x^2 + (n - 1)*x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of numbers used for basis change between certain falling factorials.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.