{"id":"n-4301181d23f6be2c","revision":0,"first_index":0,"fingerprint":"f6-9e2530a6c09e04bb","name":"theta_plus_one of Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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"],["1","2"],["1","46","3"],["1","412","618","4"],["1","3354","19023","6708","5"],["1","26904","400353","533804","67260","6"],["1","215318","7210836","23664924","12018060","645954","7"],["1","1722644","120072204","803183948","1003979935","240144408","6029254","8"],["1","13781266","1914454437","23500813784","58437904315","35251220676","4467060353","55125064","9"],["1","110250256","29768499267","627900631540","2759263704960","3311116445952","1098826105195","79382664712","496126152","10"],["1","882002190","456098589186","15816248753092","113973938509725","238891262504790","159563513913615","31632497506184","1368295767558","4410010950","11"],["1","7056017676","6927308675718","382898804672652","4299253894376250","14504318492395914","16921704907795233","6878806231002000","861522310513467","23091028919060","38808097218","12"],["1","56448141578","104670357041259","9019160963292016","152098294946427260","781068080472309594","1457217567380845823","1041424107296412792","273776930903569068","22547902408230040","383791309151283","338688849468","13"],["1","451585132808","1576733876369109","208394989873303572","5135733795050637525","38558088004051530468","108341112025948996670","123818413743941710480","57837132006077295702","10271467590101275050","573086222151584823","6306935505476436","2935303363252","14"],["1","3612681062662","23709177955456224","4749647023687451004","167510253066152985380","1783793149283294047752","7229159392042424161283","12393499114933217814584","9294633504054545350221","2972988582138823412920","368522556745536567836","14248941071062353012","102739771140310304","25288767438634","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","50","1035","29091","1028328","43755100","2175132402","123640359905","7906864428045","561707147557302","43878568842465201","3737640901174270195","344751315433466675910","34229231618209288411832"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"],["1","46","3"],["1","412","618","4"]],"recurrence":{"denominator":{"coeffs":[["-43"],["42"]]},"terms":[{"coeff":{"coeffs":[["0","0","49"],["0","0","-2058"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-84","161"],["0","3528","7742"],["0","0","-4116"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","49","-49"],["0","0","-2058","2058"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-6","-446","-559"],["252","-6048","-3066"],["0","3528","5635"],["0","0","-2058"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-42","343","-301"],["0","1764","49","-1813"],["0","0","-2058","2058"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-37","516"],["-210","-1106"],["0","588"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-301","301"],["0","294","-294"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191285,"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-b094429a591a683e"]},{"id":191286,"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-b094429a591a683e"]},{"id":191287,"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-b094429a591a683e"]},{"id":191288,"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-b094429a591a683e"]}],"region_exact":{},"incoming_edges":[{"id":"e-b094429a591a683e","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-4301181d23f6be2c","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":"7c6f879f948a3c1f653d06c29d580af31b171b5d9a37b02fb4cf22981d670340","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-2d05bfc763190112","role":"sequence","name":"Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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,         2;\n  2: 1,        46,           3;\n  3: 1,       412,         618,            4;\n  4: 1,      3354,       19023,         6708,             5;\n  5: 1,     26904,      400353,       533804,         67260,             6;\n  6: 1,    215318,     7210836,     23664924,      12018060,        645954,             7;\n  7: 1,   1722644,   120072204,    803183948,    1003979935,     240144408,       6029254,           8;\n  8: 1,  13781266,  1914454437,  23500813784,   58437904315,   35251220676,    4467060353,    55125064,         9;\n  9: 1, 110250256, 29768499267, 627900631540, 2759263704960, 3311116445952, 1098826105195, 79382664712, 496126152, 10;\n","recurrence_sentence":"The row polynomials satisfy (42*n-43)*P(n,x) = -(2058*n-49)*x^2*P(n-4,x) + ((3528*n-84)*x-(4116*n^2-7742*n-161)*x^2)*P(n-3,x) - ((2058*n-49)*x^2-(2058*n-49)*x^3)*d/dx P(n-3,x) + (252*n-6+(3528*n^2-6048*n-446)*x-(2058*n^3-5635*n^2+3066*n+559)*x^2)*P(n-2,x) + ((1764*n-42)*x-(2058*n^2-49*n-343)*x^2+(2058*n^2-1813*n-301)*x^3)*d/dx P(n-2,x) - (210*n+37-(588*n^2-1106*n+516)*x)*P(n-1,x) + ((294*n-301)*x-(294*n-301)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 46*x + 3*x^2, P(3,x) = 1 + 412*x + 618*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 46*x + 3*x^2;\nP[3, x_] := 1 + 412*x + 618*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2058*n - 49)*x^2*P[n - 4, x] + ((3528*n - 84)*x - (4116*n^2 - 7742*n - 161)*x^2)*P[n - 3, x] - ((2058*n - 49)*x^2 - (2058*n - 49)*x^3)*D[P[n - 3, x], x] + (252*n - 6 + (3528*n^2 - 6048*n - 446)*x - (2058*n^3 - 5635*n^2 + 3066*n + 559)*x^2)*P[n - 2, x] + ((1764*n - 42)*x - (2058*n^2 - 49*n - 343)*x^2 + (2058*n^2 - 1813*n - 301)*x^3)*D[P[n - 2, x], x] - (210*n + 37 - (588*n^2 - 1106*n + 516)*x)*P[n - 1, x] + ((294*n - 301)*x - (294*n - 301)*x^2)*D[P[n - 1, x], x])/(42*n - 43)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 46*x + 3*x^2; v[4] = 1 + 412*x + 618*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(2058*n - 49)*x^2*v[i - 4] + ((3528*n - 84)*x - (4116*n^2 - 7742*n - 161)*x^2)*v[i - 3] - ((2058*n - 49)*x^2 - (2058*n - 49)*x^3)*deriv(v[i - 3], x) + (252*n - 6 + (3528*n^2 - 6048*n - 446)*x - (2058*n^3 - 5635*n^2 + 3066*n + 559)*x^2)*v[i - 2] + ((1764*n - 42)*x - (2058*n^2 - 49*n - 343)*x^2 + (2058*n^2 - 1813*n - 301)*x^3)*deriv(v[i - 2], x) - (210*n + 37 - (588*n^2 - 1106*n + 516)*x)*v[i - 1] + ((294*n - 301)*x - (294*n - 301)*x^2)*deriv(v[i - 1], x))/(42*n - 43)); 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-2d05bfc763190112","steps":["\u03b8+1"],"edges":["e-b094429a591a683e"],"text":"A144445 \u2192 \u03b8+1"},"headline":"A144445 \u2192 \u03b8+1","chain":null}