{"id":"n-612add0f6ac4b3b6","revision":0,"first_index":0,"fingerprint":"f6-bfe7b886c31e21fa","name":"theta_plus_one_squared of basis_map_A013611 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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"],["2","16"],["8","96","144"],["48","704","1728","1024"],["384","6400","20160","20480","6400"],["3840","70144","259200","348160","192000","36864"],["46080","903168","3741696","6021120","4480000","1548288","200704"],["645120","13381632","60512256","110903296","100352000","47480832","11239424","1048576"],["10321920","224428032","1088630784","2204762112","2298777600","1337720832","438337536","75497472","5308416"],["185794560","4204855296","21615206400","47427092480","55157760000","37319933952","15173222400","3649044480","477757440","26214400"],["3715891200","87069818880","470147825664","1102249984000","1399574528000","1064107376640","506650755072","152253235200","28028436480","2883584000","126877696"],["81749606400","1974990274560","11126880534528","27592773074944","37679702016000","31471911567360","16939790106624","5996562874368","1387407605760","201850880000","16747855872","603979776"],["1961990553600","48707760291840","284820045299712","741351037599744","1076767680102400","972360961228800","577902036647936","232419269541888","63655382089728","11695816704000","1378906800128","94220845056","2835349504"],["51011754393600","1297793616445440","7843691020419072","21300513966391296","32629403667660800","31483566829338624","20319418186203136","9062144301596672","2831662486388736","618438800179200","92459329650688","9013794177024","516033609728","13153337344"],["1428329123020800","37154409330769920","231303491119742976","652191749426380800","1046751514984448000","1069485236347207680","740354239728975872","359898633417523200","125163655145717760","31299804856320000","5582105023086592","692523211161600","56763697070080","2762200842240","60397977600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","248","3504","53824","910208","16941056","345563136","7683784704","185236881408","4816808312832","134470970376192","4013117943054336","127539646939791360","4301365278010245120"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-4","32","144"],["8","-32","0"],["-4","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-32","64"],["0","0","32","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","16"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187566,"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-b074bfb1e8e7f053"]},{"id":187567,"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-b074bfb1e8e7f053"]},{"id":187568,"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-b074bfb1e8e7f053"]},{"id":187569,"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-b074bfb1e8e7f053"]}],"region_exact":{},"incoming_edges":[{"id":"e-b074bfb1e8e7f053","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-612add0f6ac4b3b6","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":"0d7cd10010638528b9f4d0ff5ab316fb9a83aa1c17fd9fc593ef9d53687628bf","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-dcd6be200625a597","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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":[{"id":"l-f6a60bc080a05820","kind":"fingerprint_related","node_a_id":"n-482356feab646ef2","node_b_id":"n-612add0f6ac4b3b6","fingerprint":"f6-bfe7b886c31e21fa","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bfe7b886c31e21fa"},"other_id":"n-482356feab646ef2","other":{"id":"n-482356feab646ef2","name":"theta_plus_one_squared of basis_map_A013609 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-bfe7b886c31e21fa","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","8"],["2","24","36"]],"card_rows":[["1"],["1","8"],["2","24","36"],["6","88","216","128"]],"row_sums_preview":["1","9","62","438","3364","28444","264704","2699712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-5d127e92ec7ea44a","e-8c79368adffa8681"],"text":"A132393 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A132393 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}}],"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:         2,         16;\n  2:         8,         96,         144;\n  3:        48,        704,        1728,        1024;\n  4:       384,       6400,       20160,       20480,        6400;\n  5:      3840,      70144,      259200,      348160,      192000,       36864;\n  6:     46080,     903168,     3741696,     6021120,     4480000,     1548288,      200704;\n  7:    645120,   13381632,    60512256,   110903296,   100352000,    47480832,    11239424,    1048576;\n  8:  10321920,  224428032,  1088630784,  2204762112,  2298777600,  1337720832,   438337536,   75497472,   5308416;\n  9: 185794560, 4204855296, 21615206400, 47427092480, 55157760000, 37319933952, 15173222400, 3649044480, 477757440, 26214400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-8*n+4+(32*n-32)*x-144*x^2)*P(n-2,x) + ((32*n-32)*x^2+64*x^3)*d/dx P(n-2,x) + (4*n-2+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 16*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 8*n + 4 + (32*n - 32)*x - 144*x^2)*P[n - 2, x] + ((32*n - 32)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (4*n - 2 + 16*x)*P[n - 1, x] - 16*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 8*n + 4 + (32*n - 32)*x - 144*x^2)*v[i - 2] + ((32*n - 32)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (4*n - 2 + 16*x)*v[i - 1] - 16*x^2*deriv(v[i - 1], 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-4f6e747b794bcc2e","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-38e97daa1f393e3a","e-b074bfb1e8e7f053"],"text":"A028338 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A028338 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}