A156992 → basis A007318 → basis A013609
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(3 + 2\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 6 | 4 | 10 | ||||||
| 2 | 54 | 72 | 24 | 150 | |||||
| 3 | 648 | 1296 | 864 | 192 | 3000 | ||||
| 4 | 9720 | 25920 | 25920 | 11520 | 1920 | 75000 | |||
| 5 | 174960 | 583200 | 777600 | 518400 | 172800 | 23040 | 2250000 | ||
| 6 | 3674160 | 14696640 | 24494400 | 21772800 | 10886400 | 2903040 | 322560 | 78750000 | |
| 7 | 88179840 | 411505920 | 823011840 | 914457600 | 609638400 | 243855360 | 54190080 | 5160960 | 3150000000 |
Compute more
?
Technical details
- Node id
n-511785ca0b4480e5· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-f06e51c139cfae05
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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 6, 4; 2: 54, 72, 24; 3: 648, 1296, 864, 192; 4: 9720, 25920, 25920, 11520, 1920; 5: 174960, 583200, 777600, 518400, 172800, 23040; 6: 3674160, 14696640, 24494400, 21772800, 10886400, 2903040, 322560; 7: 88179840, 411505920, 823011840, 914457600, 609638400, 243855360, 54190080, 5160960; 8: 2380855680, 12697896960, 29628426240, 39504568320, 32920473600, 17557585920, 5852528640, 1114767360, 92897280; 9: 71425670400, 428554022400, 1142810726400, 1777705574400, 1777705574400, 1185137049600, 526727577600, 150493593600, 25082265600, 1857945600;
%F FormulaThe row polynomials satisfy P(n,x) = (3*n+3+(2*n+2)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n + 3 + (2*n + 2)*x)*P[n - 1, x]];
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 + (2*n + 2)*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
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
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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
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
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Geometric basis map (elementary) polytool ? step ›
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from
theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
Triangle read by rows: row n consists of n zeros followed by n!.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Triangle read by rows: row n consists of n zeros followed by n!.?
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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-
from
theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
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from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
-
from
basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Liu–Wang theorem (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
-
-
from
basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
Related ?
Same fingerprint ?
- A120910 → basis A013610 → basis A013610
- A178822 → basis A013611 → basis A007318
- basis-map coefficients of A113214, shifted → basis A013610
- A178822 → x ↦ x+1 → basis A013609
- A038338 → basis A013610 → D
- A038338 → basis A013610
- A085880 → basis A013610
- A028326 → D → basis A013610
- A028326 → basis A013610 → D
- basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
- A028326 → basis A007318 → basis A013609
- A038208 → basis A013610
- A085881 → basis A013609 → basis A013610
- A356546 → x ↦ x+1 → basis A013609
- basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
- A085880 → basis A013609 → basis A013610
- A135065 → x ↦ x+1 → basis A013609
- A178822 → basis A013609 → basis A013610
- basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
- A038243 → basis A013611 → D
- A216973 → basis A013610
- A135838 → basis A013610
- A135089 → basis A013610
- A134346 → basis A013610
- A085881 → basis A013610
- A094305 → x ↦ x+1 → basis A013609
- A038221 → basis A013611 → x ↦ x+1
- A038208 → D → basis A013610
- A178821 → basis A013609 → basis A013610
- A120910 → x ↦ x/(x+1)
- A178820 → basis A013610
- A134059 → basis A013610
- A094305 → x ↦ x+1 → x ↦ x/(x+1)
- A356546 → basis A013609 → basis A013610
- A038325 → basis A007318 → basis A013609
- A003506 → x ↦ x+1 → basis A013609
- A196347 → basis A013610 → D
- A178822 → basis A013610
- A027466 → basis A013611 → x ↦ x/(x+1)
- A028326 → basis A013609 → basis A013610
- A038325 → basis A013609 → basis A013610
- A038243 → basis A013611
- A013610 → D → basis A013609
- A038330 → basis A013611 → basis A013610
- A178821 → basis A013610
- A156992 → basis A013611 → x ↦ x+1
- A156992 → basis A013610
- A178821 → basis A013610 → reverse
- A132200 → basis A013610
- A085880 → basis A013611 → x ↦ x+1
- A196347 → basis A013611 → x ↦ x+1
- A136572 → basis A013609 → x ↦ x+1
- A085881 → basis A013611 → basis A007318
- A178820 → basis A013609 → basis A013610
- A135065 → x ↦ x+1 → x ↦ x/(x+1)
- A038221 → basis A013610 → D
- A178821 → basis A013610 → D
- basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
- A178820 → basis A013611 → basis A007318
- A038338 → basis A013609 → basis A013610
- A156992 → basis A013610 → D
- A120910 → D → basis A013609
- A196347 → basis A013610
- basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
- A120909 → basis A013610 → x ↦ x+1
- A120910 → basis A013609
- A120910 → basis A013609 → reverse
- A356546 → basis A013610
- A085880 → D → basis A013610
- A178820 → basis A013610 → D
- A003506 → D → basis A013610
- A085880 → basis A013610 → reverse
- A038325 → basis A013610
- A131431 → x ↦ x+1 → basis A013610
- A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
- basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
- A038330 → basis A013609 → basis A013609
- A038338 → basis A013610 → reverse
- A027465 → basis A013611 → basis A013610
- basis-map coefficients of A026729 → basis A013610 → basis A013609
- A028326 → basis A013610
- basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
- A038334 → reverse
- A178820 → basis A007318 → basis A013609
- A038221 → basis A013610
- A135065 → basis A013610
- basis-map coefficients of A112227 → basis A013610 → basis A013609
- A013620: Triangle of coefficients in expansion of (2+3x)^n.
- A038328 → basis A013609 → basis A013610
- A356546 → basis A013611 → basis A007318
- A178822 → basis A013610 → D
- A013609 → basis A013610 → x ↦ x+1
- basis-map coefficients of A113214, shifted → D → basis A013610
- A120910 → basis A013609 → D
- A038220 → D
- A038221 → basis A013609 → basis A013610
- A094305 → basis A013610
- basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
- A038291 → basis A013609 → basis A013611
- A376827 → basis A013610
- A258758 → basis A013610
- A134058 → basis A013610
- A013610 → basis A013611 → basis A013609
- basis-map coefficients of A026729 → basis A007318 → basis A013610
- A013620 → D
- A124860 → basis A013610
- A028326 → basis A013611 → x ↦ x+1
- A344565 → basis A013610
- A038208 → basis A013610 → D
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
- A085880 → basis A013610 → D
- A178821 → basis A013611 → basis A007318
- A136572 → basis A013610 → basis A013609
- A038334 → D
- A085881 → basis A007318 → basis A013609
- basis-map coefficients of A127647 → basis A013610
- A038330 → basis A013609 → x ↦ x/(x+1)
- A156992 → basis A013609 → basis A013610
References
- [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
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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.