{"id":"n-8c87e3909ebe3753","revision":0,"first_index":0,"fingerprint":"f6-49673c01b4703ab3","name":"theta_plus_one of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.","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","24","3"],["1","206","207","4"],["1","1668","6510","1264","5"],["1","13370","161730","111320","6645","6"],["1","106992","3657045","6609200","1405975","32472","7"],["1","855974","79365753","327756780","174003375","14927502","152467","8"],["1","6847836","1691076072","14803407264","16636267050","3521080692","142186044","698400","9"],["1","54782738","35738576100","634976884384","1377425503530","597549512412","59990018592","1260406752","3144825","10"],["1","438261960","752564450775","26447406940960","104613358733570","83811322859328","17041506744318","907431800832","10633288797","13979800","11"],["1","3506095742","15822260468595","1082981759330900","7527172008558450","10431071573620884","3830998423701750","412896097147728","12601097738685","86617701310","61514871","12"],["1","28048766004","332430503292498","43909968097396880","523109360535389875","1200100212093547944","740744876900929012","142722255702832128","8873852411989647","164265153463300","687932871098","268433424","13"],["1","224390128106","6982471056208662","1770139184698631784","35538957318209232675","130902901177042298874","129321640738224622492","41203862902118627360","4565617395348648879","174172111142907790","2041501797140718","5362826398104","1163217757","14"],["1","1795121024928","146644345832491785","71122106075826730704","2378625119090672105505","13760787478887890909784","21017779502514275175261","10503395409333954638016","1907996483486231657595","129934722592879050160","3186813302168566235","24458961866506992","41227607926739","5010792024","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","28","418","9448","293072","11811692","597061860","36801563368","2706998829344","233584677060352","23313633405878928","2659957937648911824","343486320388018063216","49772998781619337575744"],"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"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","5"],["0","0","12"],["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","3"],["0","0","-12","9"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","-3"],["0","0","0","-9"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","-4"],["0","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","7","-3"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","3"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189619,"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-bebe1fbc9538e542"]},{"id":189620,"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-bebe1fbc9538e542"]},{"id":189621,"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-bebe1fbc9538e542"]}],"region_exact":{},"incoming_edges":[{"id":"e-bebe1fbc9538e542","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-8c87e3909ebe3753","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":"9b9c5789a1fa6a91c6c3f9746e0eb897acf96957fd7e8c8879cfb46fabfe1043","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-25ac1cc25c118491","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.","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,       24,           3;\n  3: 1,      206,         207,            4;\n  4: 1,     1668,        6510,         1264,             5;\n  5: 1,    13370,      161730,       111320,          6645,            6;\n  6: 1,   106992,     3657045,      6609200,       1405975,        32472,           7;\n  7: 1,   855974,    79365753,    327756780,     174003375,     14927502,      152467,          8;\n  8: 1,  6847836,  1691076072,  14803407264,   16636267050,   3521080692,   142186044,     698400,       9;\n  9: 1, 54782738, 35738576100, 634976884384, 1377425503530, 597549512412, 59990018592, 1260406752, 3144825, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-12*n-5)*x^2*P(n-2,x) - ((12*n+4)*x^2-(9*n+3)*x^3)*d/dx P(n-2,x) - (9*n+3)*x^3*d^2/dx^2 P(n-2,x) + (1+(6*n-4)*x)*P(n-1,x) + (7*x-3*x^2)*d/dx P(n-1,x) + 3*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 12*n - 5)*x^2*P[n - 2, x] - ((12*n + 4)*x^2 - (9*n + 3)*x^3)*D[P[n - 2, x], x] - (9*n + 3)*x^3*D[P[n - 2, x], {x, 2}] + (1 + (6*n - 4)*x)*P[n - 1, x] + (7*x - 3*x^2)*D[P[n - 1, x], x] + 3*x^2*D[P[n - 1, x], {x, 2}]];\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; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 12*n - 5)*x^2*v[i - 2] - ((12*n + 4)*x^2 - (9*n + 3)*x^3)*deriv(v[i - 2], x) - (9*n + 3)*x^3*deriv(deriv(v[i - 2], x), x) + (1 + (6*n - 4)*x)*v[i - 1] + (7*x - 3*x^2)*deriv(v[i - 1], x) + 3*x^2*deriv(deriv(v[i - 1], x), 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-25ac1cc25c118491","steps":["\u03b8+1"],"edges":["e-bebe1fbc9538e542"],"text":"A166962 \u2192 \u03b8+1"},"headline":"A166962 \u2192 \u03b8+1","chain":null}