{"id":"n-2e0b1f4c9fd7aabe","revision":0,"first_index":0,"fingerprint":"f6-4905f7f4d43a1fb8","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","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":true,"rows":[["1"],["9","2"],["81","36","3"],["729","486","81","4"],["6561","5832","1458","144","5"],["59049","65610","21870","3240","225","6"],["531441","708588","295245","58320","6075","324","7"],["4782969","7440174","3720087","918540","127575","10206","441","8"],["43046721","76527504","44641044","13226976","2296350","244944","15876","576","9"],["387420489","774840978","516560652","178564176","37200870","4960116","428652","23328","729","10"],["3486784401","7748409780","5811307335","2295825120","558013050","89282088","9644670","699840","32805","900","11"],["31381059609","76709256822","63924380685","28410835860","7891898850","1473154452","190964466","17321040","1082565","44550","1089","12"],["282429536481","753145430616","690383311398","340930030320","106540634475","22728668688","3437360388","374134464","29229255","1603800","58806","1296","13"],["2541865828329","7343167948506","7343167948506","3988881354744","1385028248175","332406779562","57453023628","7295622048","683964567","46911150","2293434","75816","1521","14"],["22876792454961","71172243193212","77103263459313","45690822790704","17451355927005","4653694913868","904885122141","131321196864","14363255907","1182160980","72243171","3184272","95823","1764","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","120","1300","14000","150000","1600000","17000000","180000000","1900000000","20000000000","210000000000","2200000000000","23000000000000","240000000000000"],"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":[["9","0"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["81","18","1"],["0","9","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185863,"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-7972dcb2788809f1"]},{"id":185864,"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-7972dcb2788809f1"]},{"id":185865,"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-7972dcb2788809f1"]},{"id":185866,"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-7972dcb2788809f1"]}],"region_exact":{},"incoming_edges":[{"id":"e-7972dcb2788809f1","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-2e0b1f4c9fd7aabe","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":"a2cb82f02c7f3ba7a5d323dcdb56bbf15204d51cd0cfb666b06224e18b019f7f","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-3dd184aba99388f7","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","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:         9,         2;\n  2:        81,        36,         3;\n  3:       729,       486,        81,         4;\n  4:      6561,      5832,      1458,       144,        5;\n  5:     59049,     65610,     21870,      3240,      225,       6;\n  6:    531441,    708588,    295245,     58320,     6075,     324,      7;\n  7:   4782969,   7440174,   3720087,    918540,   127575,   10206,    441,     8;\n  8:  43046721,  76527504,  44641044,  13226976,  2296350,  244944,  15876,   576,   9;\n  9: 387420489, 774840978, 516560652, 178564176, 37200870, 4960116, 428652, 23328, 729, 10;\n","recurrence_sentence":"The row polynomials satisfy (9+n*x)*P(n,x) = (81+(9*n+18)*x+(n+1)*x^2)*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[((81 + (9*n + 18)*x + (n + 1)*x^2)*P[n - 1, x])/(9 + n*x)]];\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] = ((81 + (9*n + 18)*x + (n + 1)*x^2)*v[i - 1])/(9 + n*x)); 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-3dd184aba99388f7","steps":["\u03b8+1"],"edges":["e-7972dcb2788809f1"],"text":"A038291 \u2192 \u03b8+1"},"headline":"A038291 \u2192 \u03b8+1","chain":null}