{"id":"n-7a174e012bdbab42","revision":0,"first_index":0,"fingerprint":"f6-5e4d568b2cdb4901","name":"theta_plus_one of basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1","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"],["1","6"],["4","30","27"],["28","234","324","108"],["280","2508","4293","2376","405"],["3640","34284","67095","48060","14175","1458"],["58240","570384","1227798","1037340","407025","74358","5103"],["1106560","11186736","25894890","24620652","11623500","2878092","357210","17496"],["24344320","252747552","620027892","645233904","348044445","105162624","17931942","1609632","59049"],["608608000","6464754720","16638061284","18610959168","11120738265","3882025602","816367734","101721744","6908733","196830"],["17041024000","184664780160","494957112192","587659101840","381171768300","148731374610","36445386159","5647183920","536755410","28540350","649539"],["528271744000","5826854328960","16174661988672","20197260605808","14020046449200","5982890978790","1650366782064","300018311208","35698663440","2673935550","114318864","2125764"],["17961239296000","201282677648640","576159352095168","751405508552160","552421306544580","253890460495980","77052589015941","15859022976720","2226316357287","209909353500","12710828691","446410440","6908733"],["664565853952000","7555226508775680","22223668076940096","30106641224810592","23257358999220060","11382663741911748","3739562405325747","850964155336152","135897907766049","15187700603790","1163001528117","58116261996","1706457051","22320522"],["26582634158080000","306196455474739200","922945242367094400","1293160321300184064","1043194264561842120","539033042073662136","189421819309720998","46859923031705784","8307920334901473","1060501050038430","96639473117187","6130837299132","257136133527","6405989814","71744535"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","61","694","9862","168712","3380248","77685136","2015161360","58250342080","1856883676480","64718860232320","2448481546483840","99932959421209600","4376865157586252800"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-25","30","-9"],["30","-18","0"],["-9","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-7","6"],["6","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187365,"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-faab9be4a6c95826"]},{"id":187366,"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-faab9be4a6c95826"]},{"id":187367,"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-faab9be4a6c95826"]},{"id":187368,"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-faab9be4a6c95826"]}],"region_exact":{},"incoming_edges":[{"id":"e-faab9be4a6c95826","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-7a174e012bdbab42","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":"ad2f48fb3f7c806a86b5cece6a6915e211de15af6b76b3601a3b0d1f2f6cacfc","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-b0592fccca597966","role":"sequence","name":"basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1","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:         1,          6;\n  2:         4,         30,          27;\n  3:        28,        234,         324,         108;\n  4:       280,       2508,        4293,        2376,         405;\n  5:      3640,      34284,       67095,       48060,       14175,       1458;\n  6:     58240,     570384,     1227798,     1037340,      407025,      74358,      5103;\n  7:   1106560,   11186736,    25894890,    24620652,    11623500,    2878092,    357210,     17496;\n  8:  24344320,  252747552,   620027892,   645233904,   348044445,  105162624,  17931942,   1609632,   59049;\n  9: 608608000, 6464754720, 16638061284, 18610959168, 11120738265, 3882025602, 816367734, 101721744, 6908733, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(18*n-30)*x+9*x^2)*P(n-2,x) + (6*n-7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*P[n - 2, x] + (6*n - 7 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*v[i - 2] + (6*n - 7 + 6*x)*v[i - 1]); 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-1a0f876fa1441737","steps":["basis A013610","\u03b8+1"],"edges":["e-6e846af4747cc041","e-faab9be4a6c95826"],"text":"basis-map coefficients of A371080, shifted \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A371080, shifted \u2192 basis A013610 \u2192 \u03b8+1","chain":null}