{"id":"n-5f48e4f1f9f7fc7a","revision":0,"first_index":1,"fingerprint":"f6-3ebf8f974508976d","name":"theta_plus_one of basis_map_A013610 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","6"],["11","42","27"],["29","186","270","108"],["76","690","1620","1404","405"],["199","2340","7695","10584","6480","1458"],["521","7524","31995","61128","58725","27702","5103"],["1364","23358","122148","300780","398925","293058","112266","17496"],["3571","70734","439560","1329804","2265975","2287602","1357398","437400","59049"],["9349","210270","1514835","5446872","11385765","14727258","11966535","5948640","1653372","196830"],["24476","616170","5051160","21070152","52317090","82882926","86087610","58436640","24977727","6101730","649539"],["64079","1785096","16411410","77968224","224578575","422263044","536386536","464518800","270503469","101367450","22084326","2125764"],["167761","5123592","52216002","278480160","913799475","1992389076","3000992652","3174159312","2354283630","1199678850","400116024","78653268","6908733"],["439204","14592246","163292760","966336264","3561120450","8844582294","15439953186","19347076800","17505135099","11345281200","5137203951","1543304664","276349320","22320522"],["1149851","41288370","503327385","3273699672","13393322865","37361391426","74274904935","107804053440","115457505867","91186615080","52450923789","21363928200","5837879385","959782446","71744535"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","80","593","4195","28756","192698","1269395","8251093","53059726","338215220","2140056773","13456968535","84166987960","523981517246"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["90","108","-162"],["-60","-72","351"],["0","0","-243"],["0","0","54"]]},"terms":[{"coeff":{"coeffs":[["60","162","108","0"],["-90","-153","108","162"],["0","-45","-297","-243"],["0","0","81","81"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-25","-150","-261","-108","0"],["0","0","-90","-324","-162"],["0","0","45","162","81"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180269,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f12ff59711d0af12"]},{"id":180270,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-f12ff59711d0af12"]},{"id":180271,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-f12ff59711d0af12"]},{"id":180272,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-f12ff59711d0af12"]}],"region_exact":{},"incoming_edges":[{"id":"e-f12ff59711d0af12","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-5f48e4f1f9f7fc7a","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":"ddc3de049994dad66fb076f3b106695b22c4bd5e8c1cf5fb40ec7673d452d308","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b362ab32ef269db1","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    4,      6;\n   3:   11,     42,      27;\n   4:   29,    186,     270,     108;\n   5:   76,    690,    1620,    1404,      405;\n   6:  199,   2340,    7695,   10584,     6480,     1458;\n   7:  521,   7524,   31995,   61128,    58725,    27702,     5103;\n   8: 1364,  23358,  122148,  300780,   398925,   293058,   112266,   17496;\n   9: 3571,  70734,  439560, 1329804,  2265975,  2287602,  1357398,  437400,   59049;\n  10: 9349, 210270, 1514835, 5446872, 11385765, 14727258, 11966535, 5948640, 1653372, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(60*n-90+(72*n-108)*x-(54*n^3-243*n^2+351*n-162)*x^2)*P(n,x) = -(90*n-60+(45*n^2+153*n-162)*x-(81*n^3-297*n^2+108*n+108)*x^2-(81*n^3-243*n^2+162*n)*x^3)*P(n-1,x) - (25+150*x-(45*n^2-90*n-261)*x^2-(162*n^2-324*n-108)*x^3-(81*n^2-162*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(90*n - 60 + (45*n^2 + 153*n - 162)*x - (81*n^3 - 297*n^2 + 108*n + 108)*x^2 - (81*n^3 - 243*n^2 + 162*n)*x^3)*P[n - 1, x] - (25 + 150*x - (45*n^2 - 90*n - 261)*x^2 - (162*n^2 - 324*n - 108)*x^3 - (81*n^2 - 162*n)*x^4)*D[P[n - 1, x], x])/(-60*n + 90 - (72*n - 108)*x + (54*n^3 - 243*n^2 + 351*n - 162)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(90*n - 60 + (45*n^2 + 153*n - 162)*x - (81*n^3 - 297*n^2 + 108*n + 108)*x^2 - (81*n^3 - 243*n^2 + 162*n)*x^3)*v[i - 1] - (25 + 150*x - (45*n^2 - 90*n - 261)*x^2 - (162*n^2 - 324*n - 108)*x^3 - (81*n^2 - 162*n)*x^4)*deriv(v[i - 1], x))/(-60*n + 90 - (72*n - 108)*x + (54*n^3 - 243*n^2 + 351*n - 162)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["basis A013610","\u03b8+1"],"edges":["e-258aa4ea5bc7ae92","e-f12ff59711d0af12"],"text":"A111125 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A111125 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}