A257612 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	32	32	4													68
3	384	576	208	8												1176
4	6144	12288	7424	1280	16											27152
5	122880	307200	261120	84480	7744	32										783456
6	2949120	8847360	9707520	4669440	906752	46592	64									27126848
7	82575360	289013760	388792320	249446400	76253184	9440256	279808	128								1095801216
8	2642411520	10569646080	16845373440	13542359040	5710086144	1180827648	96641024	1679360	256							50589024512
9	95126814720	428070666240	788759838720	762996326400	412133621760	121674792960	17691279360	979845120	10077184	512						2627443262976
10	3805072588800	19025362944000	39794717491200	45026692300800	29729111408640	11500435537920	2473206743040	259436052480	9879068672	60465152	1024					151623974601728
11	167423193907200	920827566489600	2155573621555200	2793874548326400	2183215888465920	1048967349534720	302473716695040	48668698214400	3750653558784	99274407936	362795008	2048				9624874873952256
12	8036313307545600	48217879845273600	124897702654771200	182491281358848000	164854643074007040	94867562440949760	34532520288583680	7627795731578880	936047024799744	53700442718208	995646439424	2176778240	4096			666516443992297472
13	417888291992371200	2716273897950412800	7713521389692518400	12545354765854310400	12869435842300477440	8640819862626631680	3812937441308835840	1080912625061068800	186532446590730240	17708053529886720	763753955328000	9973878620160	13060685824	8192		50002158357801885696
14	23401744351572787200	163812210461009510400	506062721602761523200	906817593623445504000	1041682333857900134400	800108077061976883200	415801873451188224000	144653676740975001600	32580506969765314560	4457874895593799680	330965026823208960	10812241918033920	99843271688192	78364147712	16384	4039720490206565777408

Technical details

Node id: n-37818242788af2ca · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-2a2e8db5a042c827

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 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:           1;
  1:           4,            2;
  2:          32,           32,            4;
  3:         384,          576,          208,            8;
  4:        6144,        12288,         7424,         1280,           16;
  5:      122880,       307200,       261120,        84480,         7744,           32;
  6:     2949120,      8847360,      9707520,      4669440,       906752,        46592,          64;
  7:    82575360,    289013760,    388792320,    249446400,     76253184,      9440256,      279808,       128;
  8:  2642411520,  10569646080,  16845373440,  13542359040,   5710086144,   1180827648,    96641024,   1679360,      256;
  9: 95126814720, 428070666240, 788759838720, 762996326400, 412133621760, 121674792960, 17691279360, 979845120, 10077184, 512;

%F Formula

The row polynomials satisfy P(n,x) = (4*n+(4*n-2)*x)*P(n-1,x) - (4*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n + (4*n - 2)*x)*P[n - 1, x] - (4*x + 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n + (4*n - 2)*x)*v[i - 1] - (4*x + 4*x^2)*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

References

[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