A121574 → reverse → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-3\,n + 11\,n x - 6\,n x^{2} - 12\,n x^{3} + 8\,n x^{4} + 6\,n^{2} - 4\,n^{2} x + 4\,n^{2} x^{2} + 8\,n^{2} x^{3} - 3\,n^{3} + 6\,n^{3} x\right)\,P_{n}(x) = \left(-40\,x^{2} - 28\,x^{3} + 16\,x^{4} + 16\,x^{5} + 13\,n x + 44\,n x^{2} + 28\,n x^{3} + 3\,n^{2} + 38\,n^{2} x + 16\,n^{2} x^{2} - 16\,n^{2} x^{3} - 3\,n^{3} - 12\,n^{3} x\right)\,P_{n-1}(x) + \left(-3\,x^{2} - 20\,x^{3} + 24\,x^{4} + 48\,x^{5} + 16\,x^{6} + 4\,n x^{2} + 36\,n x^{3} + 48\,n x^{4} + 16\,n x^{5} + 3\,n^{2} x + 24\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 3 | 2 | 5 | |||||||||||||
| 2 | 6 | 13 | 4 | 23 | ||||||||||||
| 3 | 9 | 40 | 40 | 8 | 97 | |||||||||||
| 4 | 12 | 85 | 169 | 108 | 16 | 390 | ||||||||||
| 5 | 15 | 148 | 469 | 582 | 272 | 32 | 1518 | |||||||||
| 6 | 18 | 229 | 1021 | 2001 | 1784 | 656 | 64 | 5773 | ||||||||
| 7 | 21 | 328 | 1906 | 5256 | 7328 | 5072 | 1536 | 128 | 21575 | |||||||
| 8 | 24 | 445 | 3205 | 11562 | 22601 | 24248 | 13680 | 3520 | 256 | 79541 | ||||||
| 9 | 27 | 580 | 4999 | 22458 | 57431 | 85994 | 74640 | 35488 | 7936 | 512 | 290065 | |||||
| 10 | 30 | 733 | 7369 | 39807 | 126992 | 248333 | 299436 | 217648 | 89344 | 17664 | 1024 | 1048380 | ||||
| 11 | 33 | 904 | 10396 | 65796 | 253019 | 617240 | 969624 | 974872 | 608512 | 219648 | 38912 | 2048 | 3761004 | |||
| 12 | 36 | 1093 | 14161 | 102936 | 465023 | 1368755 | 2679825 | 3500884 | 3010912 | 1645056 | 529664 | 84992 | 4096 | 13407433 | ||
| 13 | 39 | 1300 | 18745 | 154062 | 801506 | 2775470 | 6556425 | 10659790 | 11880928 | 8912832 | 4326656 | 1256960 | 184320 | 8192 | 47537225 | |
| 14 | 42 | 1525 | 24229 | 222333 | 1311176 | 5237390 | 14563755 | 28553185 | 39541360 | 38344800 | 25478528 | 11121920 | 2942976 | 397312 | 16384 | 167756915 |
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Technical details
- Node id
n-34027d3a9797cbda· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-4253ad1bd071ee12
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 ExampleTriangle begins: 0: 1; 1: 3, 2; 2: 6, 13, 4; 3: 9, 40, 40, 8; 4: 12, 85, 169, 108, 16; 5: 15, 148, 469, 582, 272, 32; 6: 18, 229, 1021, 2001, 1784, 656, 64; 7: 21, 328, 1906, 5256, 7328, 5072, 1536, 128; 8: 24, 445, 3205, 11562, 22601, 24248, 13680, 3520, 256; 9: 27, 580, 4999, 22458, 57431, 85994, 74640, 35488, 7936, 512;
%F FormulaThe row polynomials satisfy -(3*n^3-6*n^2+3*n-(6*n^3-4*n^2+11*n)*x-(4*n^2-6*n)*x^2-(8*n^2-12*n)*x^3-8*n*x^4)*P(n,x) = -(3*n^3-3*n^2+(12*n^3-38*n^2-13*n)*x-(16*n^2+44*n-40)*x^2+(16*n^2-28*n+28)*x^3-16*x^4-16*x^5)*P(n-1,x) + (3*n^2*x+(24*n^2+4*n-3)*x^2+(12*n^2+36*n-20)*x^3+(48*n+24)*x^4+(16*n+48)*x^5+16*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n^3 - 3*n^2 + (12*n^3 - 38*n^2 - 13*n)*x - (16*n^2 + 44*n - 40)*x^2 + (16*n^2 - 28*n + 28)*x^3 - 16*x^4 - 16*x^5)*P[n - 1, x] + (3*n^2*x + (24*n^2 + 4*n - 3)*x^2 + (12*n^2 + 36*n - 20)*x^3 + (48*n + 24)*x^4 + (16*n + 48)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(-3*n^3 + 6*n^2 - 3*n + (6*n^3 - 4*n^2 + 11*n)*x + (4*n^2 - 6*n)*x^2 + (8*n^2 - 12*n)*x^3 + 8*n*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(3*n^3 - 3*n^2 + (12*n^3 - 38*n^2 - 13*n)*x - (16*n^2 + 44*n - 40)*x^2 + (16*n^2 - 28*n + 28)*x^3 - 16*x^4 - 16*x^5)*v[i - 1] + (3*n^2*x + (24*n^2 + 4*n - 3)*x^2 + (12*n^2 + 36*n - 20)*x^3 + (48*n + 24)*x^4 + (16*n + 48)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(-3*n^3 + 6*n^2 - 3*n + (6*n^3 - 4*n^2 + 11*n)*x + (4*n^2 - 6*n)*x^2 + (8*n^2 - 12*n)*x^3 + 8*n*x^4)); 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
reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
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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
Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.