{"id":"n-2c6a837009b82db0","revision":0,"first_index":0,"fingerprint":"f6-97e4558a746ca129","name":"theta_plus_one of basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","4"],["12","28","9"],["48","160","111","16"],["192","832","888","304","25"],["768","4096","5904","3264","675","36"],["3072","19456","35328","27136","9380","1308","49"],["12288","90112","197376","194560","95520","22848","2303","64"],["49152","409600","1050624","1265664","809600","279936","49392","3776","81"],["196608","1835008","5394432","7684096","6073600","2770944","717024","97536","5859","100"],["786432","8126464","26935296","44302336","41753600","23761920","8200192","1655808","179388","8700","121"],["3145728","35651584","131530752","245366784","268779520","183730176","79742208","21671936","3521232","311520","12463","144"],["12582912","155189248","630718464","1315962880","1643642880","1313538048","689217536","237273088","52303680","7001280","515944","17328","169"],["50331648","671088640","2979004416","6874464256","9645260800","8830844928","5445500928","2279473152","640553472","117145600","13162864","821184","23491","196"],["201326592","2885681152","13891534848","35131490304","54716006400","56498061312","40086208512","19813367808","6804891648","1595443200","246445056","23602176","1263444","31164","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","49","335","2241","14743","95729","615071","3917825","24775207","155710257","973464047","6057963457","37547675575","231895353841"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-6","-3","0"],["18","2","0"],["-12","11","1"],["0","-6","2"]]},"terms":[{"coeff":{"coeffs":[["-24","-44","-18","-1"],["72","108","30","-2"],["-48","-28","23","9"],["0","-24","-10","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-12","-22","-12","-2"],["0","24","32","0","-8"],["0","0","12","8","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180329,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-2c6acd7aa32f25fc"]},{"id":180330,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-2c6acd7aa32f25fc"]},{"id":180331,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-2c6acd7aa32f25fc"]},{"id":180332,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-2c6acd7aa32f25fc"]}],"region_exact":{},"incoming_edges":[{"id":"e-2c6acd7aa32f25fc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-2c6a837009b82db0","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"28f6785dc86864c48306c6f635e62a6dedbe25777ce796b182df7b015e89660a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-cca535f6115f5fc0","role":"sequence","name":"basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      3,       4;\n  2:     12,      28,       9;\n  3:     48,     160,     111,      16;\n  4:    192,     832,     888,     304,      25;\n  5:    768,    4096,    5904,    3264,     675,      36;\n  6:   3072,   19456,   35328,   27136,    9380,    1308,     49;\n  7:  12288,   90112,  197376,  194560,   95520,   22848,   2303,    64;\n  8:  49152,  409600, 1050624, 1265664,  809600,  279936,  49392,  3776,   81;\n  9: 196608, 1835008, 5394432, 7684096, 6073600, 2770944, 717024, 97536, 5859, 100;\n","recurrence_sentence":"The row polynomials satisfy -(12*n^2-18*n+6+(6*n^3-11*n^2-2*n+3)*x-(2*n^3+n^2)*x^2)*P(n,x) = -(48*n^2-72*n+24+(24*n^3+28*n^2-108*n+44)*x+(10*n^3-23*n^2-30*n+18)*x^2-(6*n^3+9*n^2-2*n-1)*x^3)*P(n-1,x) + ((24*n-12)*x+(12*n^2+32*n-22)*x^2+(8*n^2-12)*x^3-(4*n^2+8*n+2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(48*n^2 - 72*n + 24 + (24*n^3 + 28*n^2 - 108*n + 44)*x + (10*n^3 - 23*n^2 - 30*n + 18)*x^2 - (6*n^3 + 9*n^2 - 2*n - 1)*x^3)*P[n - 1, x] + ((24*n - 12)*x + (12*n^2 + 32*n - 22)*x^2 + (8*n^2 - 12)*x^3 - (4*n^2 + 8*n + 2)*x^4)*D[P[n - 1, x], x])/(-12*n^2 + 18*n - 6 - (6*n^3 - 11*n^2 - 2*n + 3)*x + (2*n^3 + n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(48*n^2 - 72*n + 24 + (24*n^3 + 28*n^2 - 108*n + 44)*x + (10*n^3 - 23*n^2 - 30*n + 18)*x^2 - (6*n^3 + 9*n^2 - 2*n - 1)*x^3)*v[i - 1] + ((24*n - 12)*x + (12*n^2 + 32*n - 22)*x^2 + (8*n^2 - 12)*x^3 - (4*n^2 + 8*n + 2)*x^4)*deriv(v[i - 1], x))/(-12*n^2 + 18*n - 6 - (6*n^3 - 11*n^2 - 2*n + 3)*x + (2*n^3 + n^2)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c85abc56078671cc","steps":["basis A007318","\u03b8+1"],"edges":["e-2b8d1605acbc4584","e-2c6acd7aa32f25fc"],"text":"A232535 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A232535 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}