{"id":"n-d49f9ac2a064675e","revision":0,"first_index":0,"fingerprint":"f6-d6fc3f107f959e08","name":"theta_plus_one of basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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","2"],["10","26","12"],["100","440","471","144"],["1700","10680","18567","12496","2880"],["44200","362680","883242","943796","465380","86400"],["1635400","16601560","52264674","77316068","59689880","23301216","3628800"],["81770000","990347200","3833448360","7280428768","7720103165","4674825744","1513492848","203212800"],["5315050000","74839128000","344247474600","800348796640","1084241007165","896767596432","447430690672","123910023168","14631321600"],["435834100000","6997846596000","37321246969200","102807328581280","169943078247330","178923168803862","121433854497928","51579920121984","12491069232384","1316818944000"],["44019244100000","793949326196000","4819122933289200","15359706449269280","30015166975640330","38464409438869662","33139188998074628","19087726732083584","7064339006193984","1520895294720000","144850083840000"],["5370347780200000","107514474868112000","732034800565856400","2651369353381509760","5985005971480098860","9050860196405074680","9472873523552205235","6911379082762231520","3460166160160543920","1135310359010808960","220102873955712000","19120211066880000"],["778700428129000000","17136259016573840000","129368172653561370000","524999237948963344000","1345072349473286091500","2346583760350496911608","2894111173911122305315","2561114295447972203360","1621367504630760374640","718246587682254326400","211748077590890379264","37348591167922176000","2982752926433280000"],["132379072781930000000","3176364777525154800000","26336631011806901340000","118400832022592930520000","339568388427177141975000","671699911732766894329560","954666997647330432975594","994366331244417754112040","762564331209274978852620","426558706886648239170400","169519213840582443752640","45387906076212741660672","7344972563820742656000","542861032610856960000"],["26078677338040210000000","677636457702972055600000","6122167553918355075180000","30207603479536344195960000","95903176365689164946475000","212191367569427141975443320","341664778352750125132884738","409735574728152314546604936","369482949287621285615670960","250101630720022920733361600","125361342331356101784408320","45187707583654810807716864","11084324985255191782514688","1657297832280040194048000","114000816848279961600000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","48","1155","46323","2785698","234437598","26297628885","3791731088277","683250166093968","150453374482276668","39751107354989233335","12410857463228365602087","4520265827545549143104526","1899517636741553089017468426"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"denominator":{"coeffs":[["1"],["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["-4","-4","-1"],["8","12","4"],["-8","-14","-6"],["4","8","4"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["-2","0"],["4","2"],["-4","-4"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192303,"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-65afada163a84a5f"]},{"id":192304,"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-65afada163a84a5f"]},{"id":192305,"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-65afada163a84a5f"]},{"id":192306,"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-65afada163a84a5f"]}],"region_exact":{},"incoming_edges":[{"id":"e-65afada163a84a5f","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-d49f9ac2a064675e","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":"d68a6408ec8671e480b88d4799bd217ccf7c7efc7983e743ffd865ada2e79c69","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-6a1d0871f44aed3c","role":"sequence","name":"basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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:            2,             2;\n  2:           10,            26,             12;\n  3:          100,           440,            471,             144;\n  4:         1700,         10680,          18567,           12496,            2880;\n  5:        44200,        362680,         883242,          943796,          465380,           86400;\n  6:      1635400,      16601560,       52264674,        77316068,        59689880,        23301216,         3628800;\n  7:     81770000,     990347200,     3833448360,      7280428768,      7720103165,      4674825744,      1513492848,      203212800;\n  8:   5315050000,   74839128000,   344247474600,    800348796640,   1084241007165,    896767596432,    447430690672,   123910023168,    14631321600;\n  9: 435834100000, 6997846596000, 37321246969200, 102807328581280, 169943078247330, 178923168803862, 121433854497928, 51579920121984, 12491069232384, 1316818944000;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(2*n^6-8*n^5+14*n^4-12*n^3+4*n^2)*x+(n^6-4*n^5+6*n^4-4*n^3+n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(2*n^4-4*n^3+2*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*v[i - 1])/(n^2 - 2*n + 1)); 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-e734a631f8279507","steps":["basis A007318","\u03b8+1"],"edges":["e-0b3719baa0dde23e","e-65afada163a84a5f"],"text":"A008955 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A008955 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}