{"id":"n-49a786e4275f4fae","revision":1,"first_index":0,"fingerprint":"f6-dbbfaec00ea58ed5","name":"theta_plus_one of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+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"],["3","2"],["5","20","3"],["7","70","63","4"],["9","168","378","144","5"],["11","330","1386","1320","275","6"],["13","572","3861","6864","3575","468","7"],["15","910","9009","25740","25025","8190","735","8"],["17","1360","18564","77792","121550","74256","16660","1088","9"],["19","1938","34884","201552","461890","453492","189924","31008","1539","10"],["21","2660","61047","465120","1469650","2116296","1424430","434112","53865","2100","11"],["23","3542","100947","980628","4085950","8112468","8008462","3922512","908523","88550","2783","12"],["25","4600","159390","1922800","10214875","26744400","36402100","26150080","9734175","1771000","139150","3600","13"],["27","5850","242190","3552120","23434125","78227370","140408100","139070880","75926565","22200750","3256110","210600","4563","14"],["29","7308","356265","6243120","50075025","207583740","475047405","620470080","467063415","200300100","47213595","5700240","308763","5684","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","28","144","704","3328","15360","69632","311296","1376256","6029312","26214400","113246208","486539264","2080374784"],"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":[["1","0"],["-2","1"],["0","-4"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["-1","10","3"],["-2","-13","-13"],["0","0","0"],["0","4","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","2","2"],["0","0","8","-8"],["0","0","8","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164997,"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-5eacdf1805a7ca9d"]},{"id":164998,"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-5eacdf1805a7ca9d"]},{"id":164999,"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-5eacdf1805a7ca9d"]},{"id":165000,"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-5eacdf1805a7ca9d"]}],"region_exact":{},"incoming_edges":[{"id":"e-5eacdf1805a7ca9d","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-49a786e4275f4fae","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":"5889e265cbc8cab8c5f71aee6979f598628ebf1e1b0c7240108deb8809b8cc51","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-c30cb06ad75c5e21","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2n+1, 2k+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:  3,    2;\n  2:  5,   20,     3;\n  3:  7,   70,    63,      4;\n  4:  9,  168,   378,    144,      5;\n  5: 11,  330,  1386,   1320,    275,      6;\n  6: 13,  572,  3861,   6864,   3575,    468,      7;\n  7: 15,  910,  9009,  25740,  25025,   8190,    735,     8;\n  8: 17, 1360, 18564,  77792, 121550,  74256,  16660,  1088,    9;\n  9: 19, 1938, 34884, 201552, 461890, 453492, 189924, 31008, 1539, 10;\n","recurrence_sentence":"The row polynomials satisfy -(2*n-1-(4*n^3-4*n^2+n)*x)*P(n,x) = -(2*n+1-(4*n^3-13*n+10)*x-(12*n^3-13*n+3)*x^2)*P(n-1,x) - (4*x-(8*n^2+8*n+2)*x^2+(8*n^2+8*n-2)*x^3)*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[(-(2*n + 1 - (4*n^3 - 13*n + 10)*x - (12*n^3 - 13*n + 3)*x^2)*P[n - 1, x] - (4*x - (8*n^2 + 8*n + 2)*x^2 + (8*n^2 + 8*n - 2)*x^3)*D[P[n - 1, x], x])/(-2*n + 1 + (4*n^3 - 4*n^2 + 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] = (-(2*n + 1 - (4*n^3 - 13*n + 10)*x - (12*n^3 - 13*n + 3)*x^2)*v[i - 1] - (4*x - (8*n^2 + 8*n + 2)*x^2 + (8*n^2 + 8*n - 2)*x^3)*deriv(v[i - 1], x))/(-2*n + 1 + (4*n^3 - 4*n^2 + 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-c30cb06ad75c5e21","steps":["\u03b8+1"],"edges":["e-5eacdf1805a7ca9d"],"text":"A103327 \u2192 \u03b8+1"},"headline":"A103327 \u2192 \u03b8+1","chain":null}