{"id":"n-5ed42a17b514c663","revision":0,"first_index":0,"fingerprint":"f6-7c3b635378939853","name":"theta_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","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"],["6","12"],["36","216","108"],["216","2808","4212","864"],["1296","31752","101088","63504","6480"],["7776","332424","1927476","2569968","831060","46656"],["46656","3318408","32105160","77099040","53508600","9955224","326592"],["279936","32105160","489791772","1927009440","2408761800","979583544","112368060","2239488"],["1679616","304062984","7033287024","42619311216","86715424800","63928966824","16411003056","1216251936","15116544"],["10077696","2837343816","96713995140","864354853008","2679634565820","3215561478984","1512620992764","257903987040","12768047172","100776960"],["60466176","26201222280","1288248413400","16447026620160","74164067775000","135053582117328","103829694885000","32894053240320","3864745240200","131006111400","665127936"],["362797056","240164565192","16755941463660","298235335838400","1890040368380400","4972050376383888","5800725439114536","3024064589408640","671029505636400","55853138212200","1320905108556","4353564672"],["2176782336","2189779257096","214040184084288","5210791461977040","45204551724787200","165650365839586608","278434821077497728","220867154452782144","81368193104616960","13026978654942600","784814008309056","13138675542576","28298170368"],["13060694016","19890863030088","2696584295551572","88435800765701424","1028627078852722500","5106021544723657968","11900422229806923624","13600482548350769856","7659032317085486952","2057254157705445000","243198452105678916","10786337182206288","129290609695572","182849716224"],["78364164096","180193229732232","33612135917514408","1466759394854203104","22494898453805569080","147960084761764628328","464320887580694195856","734426057610941572224","596983998318035394672","246600141269607713880","49488776598372251976","4400278184562609312","145652588975895768","1261352608125624","1175462461440"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","360","8100","204120","5715360","176359680","5952139200","218245104000","8642506118400","367699351219200","16730320480473600","810777069438336000","41697106428257280000","2268322589697196032000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","12"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-36","0","0"],["-18","-36","-18"],["0","-18","-27"],["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-18","0","18"],["0","-9","-18","27"],["0","0","-9","9"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","6"],["9","12"],["0","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"],["0","3","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191373,"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-fc5203ebe83181fc"]},{"id":191374,"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-fc5203ebe83181fc"]},{"id":191375,"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-fc5203ebe83181fc"]},{"id":191376,"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-fc5203ebe83181fc"]}],"region_exact":{},"incoming_edges":[{"id":"e-fc5203ebe83181fc","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-5ed42a17b514c663","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":"901090885595281cfce34926baf8a4a1493e1bfde4db7774514f55dff522312a","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-61cf3a32da4a0e22","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","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-06c8fc7a44623942","kind":"fingerprint_related","node_a_id":"n-db27be2f5676b05b","node_b_id":"n-5ed42a17b514c663","fingerprint":"f6-7c3b635378939853","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7c3b635378939853"},"other_id":"n-db27be2f5676b05b","other":{"id":"n-db27be2f5676b05b","name":"theta_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.","first_index":0,"fingerprint":"f6-7c3b635378939853","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","8"],["16","96","48"]],"card_rows":[["1"],["4","8"],["16","96","48"],["64","832","1248","256"]],"row_sums_preview":["1","12","160","2400","40320","752640","15482880","348364800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c35e92273dac3e06","steps":["\u03b8+1"],"edges":["e-081dfcccae930628"],"text":"A257613 \u2192 \u03b8+1"},"headline":"A257613 \u2192 \u03b8+1","chain":null}},{"id":"l-0c6072b1b75fabcf","kind":"fingerprint_related","node_a_id":"n-ca40f10a3c367093","node_b_id":"n-5ed42a17b514c663","fingerprint":"f6-7c3b635378939853","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7c3b635378939853"},"other_id":"n-ca40f10a3c367093","other":{"id":"n-ca40f10a3c367093","name":"theta_plus_one of Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.","first_index":1,"fingerprint":"f6-7c3b635378939853","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["4","24","12"],["8","104","156","32"],["16","392","1248","784","80"]],"card_rows":[["4","24","12"],["8","104","156","32"],["16","392","1248","784","80"],["32","1368","7932","10576","3420","192"]],"row_sums_preview":["40","300","2520","23520","241920","2721600","33264000","439084800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c8a3637306e75dc","steps":["\u03b8+1"],"edges":["e-995d7d6de5709b6d"],"text":"A256890 \u2192 \u03b8+1"},"headline":"A256890 \u2192 \u03b8+1","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:        6,         12;\n  2:       36,        216,         108;\n  3:      216,       2808,        4212,          864;\n  4:     1296,      31752,      101088,        63504,          6480;\n  5:     7776,     332424,     1927476,      2569968,        831060,         46656;\n  6:    46656,    3318408,    32105160,     77099040,      53508600,       9955224,        326592;\n  7:   279936,   32105160,   489791772,   1927009440,    2408761800,     979583544,     112368060,      2239488;\n  8:  1679616,  304062984,  7033287024,  42619311216,   86715424800,   63928966824,   16411003056,   1216251936,    15116544;\n  9: 10077696, 2837343816, 96713995140, 864354853008, 2679634565820, 3215561478984, 1512620992764, 257903987040, 12768047172, 100776960;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -(18*n+36+(18*n^2+36*n)*x+(9*n^3+27*n^2+18*n)*x^2)*P(n-2,x) - ((9*n+18)*x+(9*n^2+18*n)*x^2-(9*n^2+27*n+18)*x^3)*d/dx P(n-2,x) + (9*n+12+(6*n^2+12*n+6)*x)*P(n-1,x) + ((3*n+3)*x-(3*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(18*n + 36 + (18*n^2 + 36*n)*x + (9*n^3 + 27*n^2 + 18*n)*x^2)*P[n - 2, x] - ((9*n + 18)*x + (9*n^2 + 18*n)*x^2 - (9*n^2 + 27*n + 18)*x^3)*D[P[n - 2, x], x] + (9*n + 12 + (6*n^2 + 12*n + 6)*x)*P[n - 1, x] + ((3*n + 3)*x - (3*n + 3)*x^2)*D[P[n - 1, x], x])/(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] = 6 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n + 36 + (18*n^2 + 36*n)*x + (9*n^3 + 27*n^2 + 18*n)*x^2)*v[i - 2] - ((9*n + 18)*x + (9*n^2 + 18*n)*x^2 - (9*n^2 + 27*n + 18)*x^3)*deriv(v[i - 2], x) + (9*n + 12 + (6*n^2 + 12*n + 6)*x)*v[i - 1] + ((3*n + 3)*x - (3*n + 3)*x^2)*deriv(v[i - 1], x))/(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-61cf3a32da4a0e22","steps":["\u03b8+1"],"edges":["e-fc5203ebe83181fc"],"text":"A257626 \u2192 \u03b8+1"},"headline":"A257626 \u2192 \u03b8+1","chain":null}
