A013610 → basis A008277
nonnegative coefficients
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means
Recurrence
\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 4 | 3 | 7 | ||||||
| 2 | 16 | 33 | 9 | 58 | |||||
| 3 | 64 | 279 | 189 | 27 | 559 | ||||
| 4 | 256 | 2145 | 2727 | 918 | 81 | 6127 | |||
| 5 | 1024 | 15783 | 33705 | 20115 | 4050 | 243 | 74920 | ||
| 6 | 4096 | 113553 | 384399 | 362610 | 125145 | 16767 | 729 | 1007299 | |
| 7 | 16384 | 807159 | 4184649 | 5867127 | 3090150 | 694008 | 66339 | 2187 | 14728003 |
Computed in your browser ?
Technical details
- Node id
n-ea153c5d201f990d· revision 1 · JSON- Definition source
derived:basis_map_A008277- Fingerprint
f6-b4cb782974a54903
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
OEIS matches ?
No OEIS matches.
OEIS submission text ?
Fields to paste into an OEIS submission: the example %e, the formula
%F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows.
For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched;
the matched terms were compared with the published ones, which identifies the node with the entry on a
finite prefix, not for every n. The PARI/GP program is not machine-checked here.
The atlas's own rows: n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 4, 3; 2: 16, 33, 9; 3: 64, 279, 189, 27; 4: 256, 2145, 2727, 918, 81; 5: 1024, 15783, 33705, 20115, 4050, 243; 6: 4096, 113553, 384399, 362610, 125145, 16767, 729; 7: 16384, 807159, 4184649, 5867127, 3090150, 694008, 66339, 2187; 8: 65536, 5699265, 44267967, 88826598, 67043781, 22456602, 3541482, 253692, 6561; 9: 262144, 40091463, 459777465, 1287549675, 1339180290, 627806781, 145282410, 16966746, 944784, 19683;
%F FormulaThe row polynomials satisfy P(n,x) = (4+3*x)*P(n-1,x) + 3*x*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[(4 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], 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] = (4 + 3*x)*v[i - 1] + 3*x*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)Generated with polytool c48629374597.
Proved properties
-
Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.by the cited steps:
e-240eb6cab1dce1b5 -
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).derived from the cited steps by a stated theorem:
e-240eb6cab1dce1b5 -
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.derived from the cited steps by a stated theorem:
e-240eb6cab1dce1b5
Proof chains
1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON
-
basis_map A008277
e-240eb6cab1dce1b5Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
-
premise (sequence):
Triangle of coefficients in expansion of (1+3*x)^n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-035d18f235569c81root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
basis_map A013610
e-404109e50a21afd4Geometric basis map (elementary) polytool ?
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
veronese_section r=2, j=0, shift 0
e-292206ad4ae86d2dVeronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
-
premise (sequence):
basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
parent_provers periodic_product
e-1991a148b9538b93root: theorem checkParent-row provers of polytool polytool ?
-
preserver_image mobius, shift 0
e-12303ad0e0b9f8ebAffine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
premise (sequence):
basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
-
-
premise (sequence):
basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
-
parent_provers periodic_product
e-b7eca039219a7f53root: theorem checkParent-row provers of polytool polytool ?
-
preserver_image mobius, shift 0
e-7e44b66e04c497d6Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
-
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
-
preserver_image reversal, shift 0
e-b03bb7746bf562deAffine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Cube of lower triangular normalized binomial matrix.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-2786234a932620b2root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
preserver_image reversal, shift 0
e-090c65763405a203Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle of coefficients in expansion of (1+3*x)^n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-035d18f235569c81(expanded above) -
↵ basis_map A013610
e-404109e50a21afd4(expanded above) -
parent_provers periodic_product
e-bd3566bc3465f71broot: theorem checkParent-row provers of polytool polytool ?
-
preserver_image mobius, shift 0
e-33c9dd76a325511aAffine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-a6866143cc223753root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
basis_map A013609
e-41b2234a78052e4fGeometric basis map (elementary) polytool ?
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
-
gamma_polynomial scale 1, shift 0
e-4c4f74d9ba733cffGamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
-
premise (sequence):
Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
parent_provers periodic_product
e-eab26daf7024ea74root: theorem checkParent-row provers of polytool polytool ?
-
-
premise (sequence):
Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
-
preserver_image reversal, shift 0
e-6d15baf5e54583eeAffine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-0d085d05819c04f2root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
preserver_image reversal, shift 0
e-045f9d56b54ceaf2Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-a6866143cc223753(expanded above) -
↵ basis_map A013609
e-41b2234a78052e4f(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-4c4f74d9ba733cff(expanded above) -
parent_provers periodic_product
e-d08b7f9df01d4f3croot: theorem checkParent-row provers of polytool polytool ?
-
preserver_image mobius, shift 0
e-2c323256b27180b8Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
binomial
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-317eb4ccca60c6acroot: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
preserver_image reversal, shift 0
e-033cf842aad3bdf7Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
binomial
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-317eb4ccca60c6ac(expanded above) -
basis_map A007318
e-095baa2a9dfe188fGeometric basis map (elementary) polytool ?
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
-
parent_provers periodic_product
e-13983862a411779croot: theorem checkParent-row provers of polytool polytool ?
-
preserver_image reversal, shift 0
e-989c95e5c893fce9Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Riordan array (1-x,x/(1-x)).
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
parent_provers periodic_product
e-b62074ba45057e82root: theorem checkParent-row provers of polytool polytool ?
-
-
premise (sequence):
Riordan array (1-x,x/(1-x)).
-
gamma_polynomial scale 1, shift 0
e-c76adc2a99cd3deaGamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
-
premise (sequence):
Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
preserver_image reversal, shift 0
e-0e058c428dfdfeb6Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
parent_provers periodic_product
e-d82c9b48e4b31a6broot: theorem checkParent-row provers of polytool polytool ?
-
-
premise (sequence):
Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
-
↵ parent_provers periodic_product
e-d82c9b48e4b31a6b(expanded above)
-
-
premise (sequence):
Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
-
preserver_image affine, shift 0
e-3a52de7742ed9314Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
-
↵ theorem_check liu_wang
-
premise (sequence):
binomial
-
↵ basis_map A007318
e-095baa2a9dfe188f(expanded above) -
↵ parent_provers periodic_product
e-13983862a411779c(expanded above) -
↵ preserver_image reversal, shift 0
e-989c95e5c893fce9(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-c76adc2a99cd3dea(expanded above) -
↵ preserver_image affine, shift 0
e-3a52de7742ed9314(expanded above)
-
-
premise (sequence):
binomial
-
↵ theorem_check liu_wang
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
-
parent_provers periodic_product
e-1697ba43e31b7729root: theorem checkParent-row provers of polytool polytool ?
-
basis_map A007318
e-4608a916413bc814Geometric basis map (elementary) polytool ?
-
premise (sequence):
binomial
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-317eb4ccca60c6ac(expanded above) -
↵ preserver_image reversal, shift 0
e-033cf842aad3bdf7(expanded above) -
↵ basis_map A007318
e-095baa2a9dfe188f(expanded above) -
↵ parent_provers periodic_product
e-13983862a411779c(expanded above) -
↵ preserver_image reversal, shift 0
e-989c95e5c893fce9(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-c76adc2a99cd3dea(expanded above) -
↵ preserver_image affine, shift 0
e-3a52de7742ed9314(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
binomial
-
preserver_image affine, shift 0
e-17c52917511dc051Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
binomial
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-317eb4ccca60c6ac(expanded above) -
↵ preserver_image reversal, shift 0
e-033cf842aad3bdf7(expanded above) -
↵ basis_map A007318
e-095baa2a9dfe188f(expanded above) -
↵ parent_provers periodic_product
e-13983862a411779c(expanded above) -
↵ preserver_image reversal, shift 0
e-989c95e5c893fce9(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-c76adc2a99cd3dea(expanded above) -
↵ preserver_image affine, shift 0
e-3a52de7742ed9314(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
binomial
-
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
-
↵ parent_provers periodic_product
e-d08b7f9df01d4f3c(expanded above) -
↵ preserver_image mobius, shift 0
e-2c323256b27180b8(expanded above)
-
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
-
↵ theorem_check liu_wang
-
premise (sequence):
Triangle of coefficients in expansion of (1+3*x)^n.
-
parent_provers periodic_product
e-58678fea8a24dd74root: theorem checkParent-row provers of polytool polytool ?
-
basis_map A007318
e-7b97c40e19745577Geometric basis map (elementary) polytool ?
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-0d085d05819c04f2(expanded above) -
↵ preserver_image reversal, shift 0
e-045f9d56b54ceaf2(expanded above) -
↵ parent_provers periodic_product
e-1697ba43e31b7729(expanded above) -
↵ basis_map A007318
e-4608a916413bc814(expanded above) -
↵ preserver_image affine, shift 0
e-17c52917511dc051(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
-
preserver_image affine, shift 0
e-4b09f87a5a93f62bAffine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-0d085d05819c04f2(expanded above) -
↵ preserver_image reversal, shift 0
e-045f9d56b54ceaf2(expanded above) -
↵ parent_provers periodic_product
e-1697ba43e31b7729(expanded above) -
↵ basis_map A007318
e-4608a916413bc814(expanded above) -
↵ preserver_image affine, shift 0
e-17c52917511dc051(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
-
-
premise (sequence):
Cube of lower triangular normalized binomial matrix.
-
↵ parent_provers periodic_product
e-bd3566bc3465f71b(expanded above) -
↵ preserver_image mobius, shift 0
e-33c9dd76a325511a(expanded above)
-
-
premise (sequence):
Triangle of coefficients in expansion of (1+3*x)^n.
Related sequences ?
No candidate equalities or related sequences.
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
- [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
- [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