A030528 → basis A013611 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(16 + 32\,x - 4\,n - 16\,n^{2} - 32\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(80 + 64\,x - 232\,n - 576\,n x - 384\,n x^{2} + 88\,n^{2} + 128\,n^{2} x - 8\,n^{3}\right)\,P_{n-1}(x) + \left(10 + 48\,x + 32\,x^{2} - 25\,n - 160\,n x - 272\,n x^{2} - 128\,n x^{3} + 5\,n^{2} + 24\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | ||||||||||||||
| 1 | 4 | 4 | |||||||||||||
| 2 | 16 | 32 | 48 | ||||||||||||
| 3 | 40 | 192 | 192 | 424 | |||||||||||
| 4 | 88 | 672 | 1536 | 1024 | 3320 | ||||||||||
| 5 | 180 | 1888 | 6912 | 10240 | 5120 | 24340 | |||||||||
| 6 | 352 | 4704 | 23808 | 56320 | 61440 | 24576 | 171200 | ||||||||
| 7 | 668 | 10848 | 70272 | 230400 | 399360 | 344064 | 114688 | 1170300 | |||||||
| 8 | 1240 | 23712 | 187392 | 788480 | 1894400 | 2580480 | 1835008 | 524288 | 7835000 | ||||||
| 9 | 2264 | 49824 | 465024 | 2393088 | 7388160 | 13934592 | 15597568 | 9437184 | 2359296 | 51627000 | |||||
| 10 | 4080 | 101568 | 1093632 | 6661120 | 25190400 | 61071360 | 94502912 | 89653248 | 47185920 | 10485760 | 335950000 | ||||
| 11 | 7276 | 202144 | 2467008 | 17367040 | 77849600 | 231383040 | 460128256 | 602406912 | 495452160 | 230686720 | 46137344 | 2164087500 | |||
| 12 | 12864 | 394560 | 5382912 | 43018240 | 223180800 | 787046400 | 1919418368 | 3227516928 | 3656908800 | 2652897280 | 1107296256 | 201326592 | 13824400000 | ||
| 13 | 22580 | 757824 | 11430144 | 102251520 | 602956800 | 2463375360 | 7132217344 | 14707851264 | 21387018240 | 21338521600 | 13841203200 | 5234491392 | 872415232 | 87694512500 | |
| 14 | 39392 | 1435936 | 23727360 | 234939392 | 1552486400 | 7215882240 | 24220270592 | 59314274304 | 105753083904 | 135297761280 | 120510742528 | 70665633792 | 24427626496 | 3758096384 | 552976000000 |
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Technical details
- Node id
n-d5e0f1297a07d43e· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-492255dbddbd3519
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 0; 1: 4; 2: 16, 32; 3: 40, 192, 192; 4: 88, 672, 1536, 1024; 5: 180, 1888, 6912, 10240, 5120; 6: 352, 4704, 23808, 56320, 61440, 24576; 7: 668, 10848, 70272, 230400, 399360, 344064, 114688; 8: 1240, 23712, 187392, 788480, 1894400, 2580480, 1835008, 524288; 9: 2264, 49824, 465024, 2393088, 7388160, 13934592, 15597568, 9437184, 2359296;
%F FormulaThe row polynomials satisfy (4*n^3-16*n^2-4*n+16-(32*n^2-32)*x)*P(n,x) = -(8*n^3-88*n^2+232*n-80-(128*n^2-576*n+64)*x+384*n*x^2)*P(n-1,x) + (5*n^2-25*n+10+(24*n^2-160*n+48)*x+(16*n^2-272*n+32)*x^2-128*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 88*n^2 + 232*n - 80 - (128*n^2 - 576*n + 64)*x + 384*n*x^2)*P[n - 1, x] + (5*n^2 - 25*n + 10 + (24*n^2 - 160*n + 48)*x + (16*n^2 - 272*n + 32)*x^2 - 128*n*x^3)*D[P[n - 1, x], x])/(4*n^3 - 16*n^2 - 4*n + 16 - (32*n^2 - 32)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 88*n^2 + 232*n - 80 - (128*n^2 - 576*n + 64)*x + 384*n*x^2)*v[i - 1] + (5*n^2 - 25*n + 10 + (24*n^2 - 160*n + 48)*x + (16*n^2 - 272*n + 32)*x^2 - 128*n*x^3)*deriv(v[i - 1], x))/(4*n^3 - 16*n^2 - 4*n + 16 - (32*n^2 - 32)*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 (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
delannoy?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
-
from
delannoy?
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
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from
delannoy?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
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from
delannoy?
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from
Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
- ↑ Ma–Wang interval theorem (above)
-
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
Geometric basis map (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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from
basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.