{"id":"n-d705b9f7d1cd3e13","revision":0,"first_index":0,"fingerprint":"f6-c65accfe82ad099a","name":"one_plus_derivative of basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["15","8","1"],["39","37","11","1"],["96","134","67","14","1"],["228","424","305","105","17","1"],["528","1232","1168","572","151","20","1"],["1200","3376","3992","2560","955","205","23","1"],["2688","8864","12592","10032","4880","1474","267","26","1"],["5952","22528","37424","35728","21532","8456","2149","337","29","1"],["13056","55808","106240","118400","85344","41440","13664","3000","415","32","1"],["28416","135424","290816","370944","311712","180768","73584","20928","4047","501","35","1"],["61440","323072","772864","1111040","1067520","720960","350112","122784","30720","5310","595","38","1"],["132096","759808","2004224","3207424","3470720","2678016","1513248","632544","194964","43560","6809","697","41","1"],["282624","1765376","5091328","8979456","10810624","9388544","6057216","2944128","1080816","297264","60016","8564","807","44","1"],["602112","4059136","12707840","24494080","32484608","31373056","22770176","12629760","5388240","1764048","438152","80704","10595","925","47","1"],["1277952","9248768","31240192","65343488","94674944","100685312","81233152","50665472","24655488","9376224","2770768","627536","106288","12922","1051","50","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["24","88","312","1080","3672","12312","40824","134136","437400","1417176","4566456","14644152","46766808","148803480","471919608"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["15","8","1"]],"recurrence":{"denominator":{"coeffs":[["7","6","1"],["3","1","0"]]},"terms":[{"coeff":{"coeffs":[["20","24","9","1"],["6","5","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175889,"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-d014d4beadefa636"]},{"id":175890,"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-d014d4beadefa636"]},{"id":175891,"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-d014d4beadefa636"]},{"id":175892,"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-d014d4beadefa636"]}],"region_exact":{},"incoming_edges":[{"id":"e-d014d4beadefa636","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-d705b9f7d1cd3e13","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"8b94efa65fdb76493c4fc53292eae24c06266634e360d0dfe00a165c440f9363","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-39578f69b35e8349","role":"sequence","name":"basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:    15,      8,      1;\n  1:    39,     37,     11,      1;\n  2:    96,    134,     67,     14,      1;\n  3:   228,    424,    305,    105,     17,      1;\n  4:   528,   1232,   1168,    572,    151,     20,     1;\n  5:  1200,   3376,   3992,   2560,    955,    205,    23,     1;\n  6:  2688,   8864,  12592,  10032,   4880,   1474,   267,    26,    1;\n  7:  5952,  22528,  37424,  35728,  21532,   8456,  2149,   337,   29,   1;\n  8: 13056,  55808, 106240, 118400,  85344,  41440, 13664,  3000,  415,  32,  1;\n  9: 28416, 135424, 290816, 370944, 311712, 180768, 73584, 20928, 4047, 501, 35, 1;\n","recurrence_sentence":"The row polynomials satisfy (3*n+7+(n+6)*x+x^2)*P(n,x) = (6*n+20+(5*n+24)*x+(n+9)*x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = 15 + 8*x + x^2.","mathematica":"P[0, x_] := 15 + 8*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 20 + (5*n + 24)*x + (n + 9)*x^2 + x^3)*P[n - 1, x])/(3*n + 7 + (n + 6)*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 15 + 8*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 20 + (5*n + 24)*x + (n + 9)*x^2 + x^3)*v[i - 1])/(3*n + 7 + (n + 6)*x + 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-f48bed27dbc29c5f","steps":["x \u21a6 x+1","D+1"],"edges":["e-3fc4d53dea6f488e","e-d014d4beadefa636"],"text":"A391533 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A391533 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}