A038208 → D → basis A013611
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | ||||||||||||||
| 1 | 2 | 2 | |||||||||||||
| 2 | 16 | 32 | 48 | ||||||||||||
| 3 | 96 | 384 | 384 | 864 | |||||||||||
| 4 | 512 | 3072 | 6144 | 4096 | 13824 | ||||||||||
| 5 | 2560 | 20480 | 61440 | 81920 | 40960 | 207360 | |||||||||
| 6 | 12288 | 122880 | 491520 | 983040 | 983040 | 393216 | 2985984 | ||||||||
| 7 | 57344 | 688128 | 3440640 | 9175040 | 13762560 | 11010048 | 3670016 | 41803776 | |||||||
| 8 | 262144 | 3670016 | 22020096 | 73400320 | 146800640 | 176160768 | 117440512 | 33554432 | 573308928 | ||||||
| 9 | 1179648 | 18874368 | 132120576 | 528482304 | 1321205760 | 2113929216 | 2113929216 | 1207959552 | 301989888 | 7739670528 | |||||
| 10 | 5242880 | 94371840 | 754974720 | 3523215360 | 10569646080 | 21139292160 | 28185722880 | 24159191040 | 12079595520 | 2684354560 | 103195607040 | ||||
| 11 | 23068672 | 461373440 | 4152360960 | 22145925120 | 77510737920 | 186025771008 | 310042951680 | 354334801920 | 265751101440 | 118111600640 | 23622320128 | 1362182012928 | |||
| 12 | 100663296 | 2214592512 | 22145925120 | 132875550720 | 531502202880 | 1488206168064 | 2976412336128 | 4252017623040 | 4252017623040 | 2834678415360 | 1133871366144 | 206158430208 | 17832200896512 | ||
| 13 | 436207616 | 10468982784 | 115158810624 | 767725404160 | 3454764318720 | 11055245819904 | 25795573579776 | 44220983279616 | 55276229099520 | 49134425866240 | 29480655519744 | 10720238370816 | 1786706395136 | 231818611654656 | |
| 14 | 1879048192 | 48855252992 | 586263035904 | 4299262263296 | 21496311316480 | 77386720739328 | 206364588638208 | 412729177276416 | 619093765914624 | 687881962127360 | 550305569701888 | 300166674382848 | 100055558127616 | 15393162788864 | 2995809750614016 |
Compute more
?
Technical details
- Node id
n-d5693ca6982ed0eb· revision 0 · JSON- Definition source
derived:basis_map_A013611- Fingerprint
f6-11aec07ce60b6d61
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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 0; 1: 2; 2: 16, 32; 3: 96, 384, 384; 4: 512, 3072, 6144, 4096; 5: 2560, 20480, 61440, 81920, 40960; 6: 12288, 122880, 491520, 983040, 983040, 393216; 7: 57344, 688128, 3440640, 9175040, 13762560, 11010048, 3670016; 8: 262144, 3670016, 22020096, 73400320, 146800640, 176160768, 117440512, 33554432; 9: 1179648, 18874368, 132120576, 528482304, 1321205760, 2113929216, 2113929216, 1207959552, 301989888;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (4*n+8*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 8*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 8*n*x)*v[i - 1])/(n - 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
derived from Geometric basis map (elementary)
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
-
Geometric basis map (elementary) polytool ? step ›
-
from
derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.?
-
Geometric basis map (elementary) polytool ? step ›
-
from
derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
-
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
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
-
from
Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.?
-
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
-
from
Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.?
-
-
from
derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
-
Geometric basis map (elementary) polytool ? step ›
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^i.?
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^i.?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Iterated real-rootedness preservers (above)
-
Geometric basis map (elementary) polytool ? step ›
-
from
binomial?
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Riordan array (1-x,x/(1-x)).?
-
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
-
from
Riordan array (1-x,x/(1-x)).?
-
Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
-
from
binomial?
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Gamma criterion (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
binomial?
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^i.?
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle whose (i,j)-th entry is binomial(i,j)*2^i.?
-
-
from
derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.?
Related ?
Same fingerprint ?
- A038221 → basis A013611 → D
- A038338 → x ↦ x+1 → reverse
- A156992 → basis A013611 → basis A013610
- A178821 → basis A013611 → basis A013610
- A003506 → D → basis A013611
- A178822 → basis A013611
- A178821 → basis A007318
- basis-map coefficients of A026729 → basis A013609 → reverse
- A085881 → basis A013611 → basis A013610
- image under θ+1 → θ+1 → D → basis A013609
- A124860 → x ↦ x/(x+1)
- basis-map coefficients of A026729 → basis A007318 → basis A013611
- A028326 → basis A007318
- A344565 → x ↦ x/(x+1)
- A135065 → x ↦ x+1
- A196347 → basis A013609 → basis A007318
- basis-map coefficients of A113214, shifted → D → basis A007318
- A120909 → basis A013610 → basis A013610
- A135089 → basis A013611
- basis-map coefficients of A026729, shifted → D → basis A013609
- A130123 → basis A013609 → D
- A135838 → x ↦ x+1
- A159854 → x ↦ x/(x+1)
- basis-map coefficients of A113214, shifted → D → basis A013611
- A156992 → basis A007318 → D
- A356546 → x ↦ x/(x+1)
- A129186 → basis A013609
- A376827 → basis A007318
- A134346 → x ↦ x/(x+1)
- A038221 → basis A013611 → basis A013610
- A027465 → D → basis A013609
- A038325 → basis A013609 → basis A007318
- basis-map coefficients of A112227 → θ+1 → basis A013609
- A135065 → x ↦ x+1 → reverse
- A136572 → basis A013609 → D
- A038338 → basis A013611
- A344565 → basis A013611
- A216973 → basis A013611
- A258758 → basis A007318
- A178822 → x ↦ x+1 → reverse
- A178822 → x ↦ x+1 → D
- A027465 → basis A013611 → basis A013611
- A027466 → basis A013611
- A085880 → basis A013611 → D
- image under θ+1 → (θ+1)² → basis A013609
- A013609 → D
- A136572 → (θ+1)² → basis A013609
- A178821 → basis A013609 → basis A013611
- A178820 → basis A013611 → reverse
- A094305 → x ↦ x+1 → D
- A038338 → x ↦ x+1
- A038208 → basis A013611 → D
- A156992 → basis A013611
- A131431 → x ↦ x+1 → basis A007318
- basis-map coefficients of A127647 → basis A007318
- basis-map coefficients of A026729 → basis A013609 → D
- A120909 → basis A013610 → D
- A356546 → basis A013611 → D
- A038325 → x ↦ x/(x+1)
- basis-map coefficients of A026729, shifted → basis A013609
- A178821 → basis A013611 → D
- A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
- A028326 → basis A007318 → reverse
- A085881 → basis A007318
- A178821 → basis A013611 → reverse
- A028326 → basis A013611 → basis A013610
- A085880 → D → basis A007318
- basis-map coefficients of A026729, shifted → basis A013609 → D
- A120910 → D
- basis-map coefficients of A127647 → basis A013611
- A085880 → basis A013611 → basis A013610
- A038208 → basis A007318
- basis-map coefficients of A112227 → basis A013609 → basis A013610
- A136572 → basis A013609
- A094305 → x ↦ x/(x+1)
- A038330 → basis A013609
- A156992 → basis A013609 → basis A007318
- A013609: Triangle of coefficients in expansion of (1+2*x)^n.
- A094305 → x ↦ x+1
- A038325 → basis A007318
- basis-map coefficients of A112227 → basis A013609
- A085881 → x ↦ x/(x+1)
- A028326 → basis A013611 → D
- basis-map coefficients of A112227 → x ↦ x+1 → basis A007318
- A028326 → basis A007318 → D
- A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
- A013609 → basis A013610 → D
- A003506 → x ↦ x/(x+1)
- A132440 → basis A013609
- A135838 → x ↦ x/(x+1)
- A085880 → basis A013609 → basis A013611
- A196347 → basis A013611
- A178821 → x ↦ x/(x+1)
- A135089 → basis A007318
- A178822 → basis A013609 → x ↦ x+1
- basis-map coefficients of A112227 → (θ+1)² → basis A013609
- basis-map coefficients of A113214, shifted → x ↦ x+1
- basis-map coefficients of A127647 → x ↦ x/(x+1)
- A134346 → basis A013611
- A134058 → basis A013611
- A085880 → D → basis A013611
- A132200 → basis A013611
- A038208 → basis A007318 → D
- A376827 → basis A013611
- basis-map coefficients of A026729 → basis A013609 → basis A013610
- A124860 → x ↦ x+1
- A028326 → x ↦ x/(x+1)
- A136572 → D → basis A013609
- A178822 → basis A013611 → basis A013610
- A258758 → basis A013611
- A132200 → basis A007318
- A038330 → basis A013609 → D
- A085880 → basis A013611
- A178820 → x ↦ x/(x+1)
- A196347 → basis A013611 → basis A013610
- A178820 → basis A007318
- basis-map coefficients of A026729 → (θ+1)² → basis A013609
- A356546 → basis A013611 → reverse
- A135838 → basis A013611
- basis-map coefficients of A030528, shifted → D → basis A013609
- A085881 → basis A013611
- A178822 → x ↦ x+1
- A136521 → basis A013609
- A156992 → x ↦ x/(x+1)
- A356546 → basis A013611
- A027466 → basis A013611 → D
- A120910 → basis A013610 → basis A013611
- A038325 → basis A013611
- A356546 → x ↦ x+1
- A196347 → x ↦ x+1
- A038338 → basis A013609 → basis A007318
- A376827 → x ↦ x/(x+1)
- A124860 → basis A013611
- A038208 → x ↦ x/(x+1)
- A094305 → basis A013611
- basis-map coefficients of A026729, shifted → θ+1 → basis A013609
- A356546 → x ↦ x+1 → reverse
- A094305 → x ↦ x+1 → reverse
- A028326 → basis A013609 → x ↦ x+1
- basis-map coefficients of A026729, shifted → basis A013609 → basis A013610
- A344565 → basis A007318
- A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
- A131431 → θ+1 → basis A013609
- A147716 → basis A013610 → basis A013610
- A130123 → D → basis A013609
- A356546 → x ↦ x+1 → D
- A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
- A178821 → basis A007318 → reverse
- A196347 → basis A013611 → D
- A028326 → basis A013611
- A013609 → D → D
- A134059 → basis A013611
- A085881 → basis A007318 → reverse
- Veronese parent of A098172 → x ↦ x/(x+1)
- basis-map coefficients of A026729, shifted → basis A007318 → x ↦ x+1
- A038325 → basis A007318 → D
- A085880 → basis A013609 → x ↦ x+1
- A085880 → x ↦ x/(x+1)
- A131431 → x ↦ x+1 → x ↦ x/(x+1)
- A085881 → basis A013611 → D
- A178822 → basis A013611 → D
- A178820 → basis A013611 → D
- A135065 → x ↦ x+1 → D
- A178820 → basis A007318 → reverse
- A196347 → x ↦ x+1 → D
- basis-map coefficients of A030528, shifted → (θ+1)² → basis A013609
- A130123 → basis A013609
- basis-map coefficients of A030528, shifted → basis A013609 → D
- A135065 → basis A013611
- A038330 → basis A013611 → x ↦ x/(x+1)
- A178821 → basis A007318 → D
- A131431 → basis A013609
- A013609 → basis A013610
- A028326 → basis A013609 → basis A013611
- A178821 → basis A013611
- basis-map coefficients of A030528, shifted → basis A013609 → basis A013610
- A134309 → basis A013609
- basis-map coefficients of A112227 → x ↦ x+1 → basis A013611
- A038338 → x ↦ x+1 → D
- A131431 → x ↦ x+1 → basis A013611
- basis-map coefficients of A113214, shifted → basis A013609 → x ↦ x+1
- A038325 → basis A007318 → reverse
- A130123 → θ+1 → basis A013609
- A038221 → basis A013609 → basis A013611
- A038330 → basis A013611 → basis A007318
- A131436 → basis A013609
- A136572 → basis A013609 → basis A013610
- A038221 → basis A013611
- basis-map coefficients of A113214, shifted → basis A013609 → basis A013611
- A038221 → x ↦ x+1
- basis-map coefficients of A026729, shifted → basis A007318 → basis A013611
- A120909 → D
- A156992 → basis A007318
- A038329 → x ↦ x+1 → basis A013610
- basis-map coefficients of A030528, shifted → basis A013609
- basis-map coefficients of A026729 → basis A013609
- A038291 → basis A013609 → basis A013610
- basis-map coefficients of A030528, shifted → θ+1 → basis A013609
- A178822 → basis A013609 → basis A013611
- A178822 → x ↦ x/(x+1)
- basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
- A120910 → D → D
- A120909 → basis A013610 → reverse
- basis-map coefficients of A026729, shifted → (θ+1)² → basis A013609
- basis-map coefficients of A021012 → basis A013609
- Veronese parent of A098172 → x ↦ x/(x+1)
- A134346 → x ↦ x+1
- A085880 → x ↦ x+1 → reverse
- A356546 → basis A013611 → basis A013610
- A127648 → D → basis A013609
- A003506 → D → x ↦ x+1
- basis-map coefficients of A021012 → θ+1 → basis A013609
- A136572 → θ+1 → basis A013609
- A356546 → basis A013609 → basis A013611
- A038221 → x ↦ x+1 → D
- basis-map coefficients of A113214, shifted → basis A013611
- A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
- image under θ+1 → basis A013609
- A156992 → basis A013611 → D
- A013609 → basis A013610 → basis A013610
- A038208 → basis A013611
- A178820 → basis A013609 → basis A013611
- A013609 → D → basis A013610
- A003506 → x ↦ x+1
- Veronese parent of A098172 → x ↦ x/(x+1)
- A178820 → basis A013611
- A085881 → basis A007318 → D
- A178820 → basis A013611 → basis A013610
- A085881 → basis A013611 → reverse
- basis-map coefficients of A112227 → basis A013609 → D
- A132440 → θ+1 → basis A013609
- A178820 → basis A007318 → D
- A134059 → basis A007318
- A131431 → D → basis A013609
- A038330 → basis A013609 → reverse
- A134058 → basis A007318
- A085880 → x ↦ x+1
- A038338 → x ↦ x/(x+1)
- A135065 → x ↦ x/(x+1)
- A120910 → basis A007318 → basis A013609
- A120909 → basis A013610
- basis-map coefficients of A026729 → D → basis A013609
References
- [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
- [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
- [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
- [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
- [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.