{"id":"n-b265d150dcab88bb","revision":0,"first_index":0,"fingerprint":"f6-8d019252059b0315","name":"theta_plus_one of basis_map_A007318 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["3","4"],["7","12"],["25","72","36"],["81","280","180"],["331","1500","1620","480"],["1303","6804","8820","3360"],["5937","37072","63000","40320","8400"],["26785","187632","367416","282240","75600"],["133651","1068660","2502360","2520000","1134000","181440"],["669351","5892700","15479640","17962560","9702000","1995840"],["3609673","35283864","105797340","146805120","103950000","35925120","4656960"],["19674097","208837512","689445900","1073255040","875674800","363242880","60540480"],["113525595","1313920972","4816247436","8557829280","8349541200","4540536000","1271350080","138378240"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","19","133","541","3931","20287","154729","939673","7540111","51702091","436028077","3290670709","29101328803"],"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":[["8","7","0"],["0","4","1"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-8","-31","-28","-4"],["16","30","11","-2"],["0","12","17","4"],["0","0","2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-16","-60","-80","-44","-8"],["0","-16","-44","-40","-12"],["0","0","-4","-8","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176386,"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-059a32232f543eab"]},{"id":176387,"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-059a32232f543eab"]},{"id":176388,"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-059a32232f543eab"]}],"region_exact":{},"incoming_edges":[{"id":"e-059a32232f543eab","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-b265d150dcab88bb","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":"ef169a963c864f197c993bbf9a64511110826c8b1a6c792f00a84dfa148b91b0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-87f3394ac59b064b","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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;\n  2:     3,      4;\n  3:     7,     12;\n  4:    25,     72,     36;\n  5:    81,    280,    180;\n  6:   331,   1500,   1620,    480;\n  7:  1303,   6804,   8820,   3360;\n  8:  5937,  37072,  63000,  40320,  8400;\n  9: 26785, 187632, 367416, 282240, 75600;\n","recurrence_sentence":"The row polynomials satisfy (8+(4*n+7)*x+(n^2+n)*x^2)*P(n,x) = (16*n-8+(12*n^2+30*n-31)*x+(2*n^3+17*n^2+11*n-28)*x^2+(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (16+(16*n+60)*x+(4*n^2+44*n+80)*x^2+(8*n^2+40*n+44)*x^3+(4*n^2+12*n+8)*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[((16*n - 8 + (12*n^2 + 30*n - 31)*x + (2*n^3 + 17*n^2 + 11*n - 28)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (16 + (16*n + 60)*x + (4*n^2 + 44*n + 80)*x^2 + (8*n^2 + 40*n + 44)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(8 + (4*n + 7)*x + (n^2 + n)*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] = ((16*n - 8 + (12*n^2 + 30*n - 31)*x + (2*n^3 + 17*n^2 + 11*n - 28)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (16 + (16*n + 60)*x + (4*n^2 + 44*n + 80)*x^2 + (8*n^2 + 40*n + 44)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(8 + (4*n + 7)*x + (n^2 + n)*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-c914b69838ac70ec","steps":["basis A007318","\u03b8+1"],"edges":["e-6629899d205ea17b","e-059a32232f543eab"],"text":"A059344 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A059344 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}