{"id":"n-e8422031461468ed","revision":0,"first_index":0,"fingerprint":"f6-cfe1d0f41456d01a","name":"theta_plus_one_squared of basis_map_A013609 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)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","8"],["8","80","108"],["48","800","2448","1920"],["384","9088","44784","76288","42000"],["3840","118528","815904","2195968","2565600","1088640"],["46080","1760256","15657984","58262528","106273600","94341888","32598720"],["645120","29435904","322186752","1539422208","3854745600","5299670016","3795036672","1107025920"],["10321920","548388864","7141911552","41814048768","133836345600","251319730176","277123779072","166415892480","42032390400"],["185794560","11274780672","170506156032","1184341753856","4630425561600","11076382623744","16618746647552","15302044090368","7917666900480","1764322560000"],["3715891200","253736386560","4374116868096","35205473173504","162928802867200","474904539205632","905270056434688","1130873368477696","894286020980736","406732552704000","81123551308800"],["81749606400","6206470225920","120208659628032","1101121135968256","5894792840089600","20301833859932160","47064650732818432","74539742916640768","79787279954718720","55318502432563200","22454866457210880","4054836680294400"],["1961990553600","163997212999680","3527377499455488","36257304317263872","220618109809868800","877715090365906944","2400671993151172608","4616674886273204224","6254505376841428992","5858773537291468800","3617944640369528832","1326580835802808320","218904863559782400"],["51011754393600","4656325648711680","110161501448306688","1256234578867126272","8568672754842828800","38705096255824134144","122150859916276932608","276812126187669684224","454766097444853358592","538408690786950348800","448522359655741612032","249773758017324318720","83536026886816358400","12693891496366080000"],["1428329123020800","141395657112944640","3650265606871056384","45750072347316977664","345917629727512985600","1750042688579654123520","6265048652458827972608","16366114469390528282624","31651829477594444021760","45393183171691500339200","47738249926380100927488","35817713725997864189952","18168420667178786734080","5587058006691530342400","786891743269632000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","196","5216","172544","6788480","308941056","15948168192","920242808832","58675896868864","4095955934298112","310644123889696768","25432339974485442560","2235308682820384194560","209920013320900805918720"],"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","8"]],"recurrence":{"denominator":{"coeffs":[["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","48","-324"],["-16","-176","1080"],["20","192","-1008"],["-8","-64","288"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-48","-144"],["0","0","176","480"],["0","0","-192","-448"],["0","0","64","128"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","24"],["-12","-64"],["8","32"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-24"],["0","0","64"],["0","0","-32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190640,"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-61db0af244b23600"]},{"id":190641,"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-61db0af244b23600"]},{"id":190642,"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-61db0af244b23600"]},{"id":190643,"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-61db0af244b23600"]}],"region_exact":{},"incoming_edges":[{"id":"e-61db0af244b23600","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-e8422031461468ed","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":"5c1e2458a5439ec9197492287aca208b9ada32c74597503fb5a8efe133761060","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-a6229b8a5ae7f968","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)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":[],"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,           8;\n  2:         8,          80,          108;\n  3:        48,         800,         2448,          1920;\n  4:       384,        9088,        44784,         76288,         42000;\n  5:      3840,      118528,       815904,       2195968,       2565600,        1088640;\n  6:     46080,     1760256,     15657984,      58262528,     106273600,       94341888,       32598720;\n  7:    645120,    29435904,    322186752,    1539422208,    3854745600,     5299670016,     3795036672,     1107025920;\n  8:  10321920,   548388864,   7141911552,   41814048768,  133836345600,   251319730176,   277123779072,   166415892480,   42032390400;\n  9: 185794560, 11274780672, 170506156032, 1184341753856, 4630425561600, 11076382623744, 16618746647552, 15302044090368, 7917666900480, 1764322560000;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(64*n^3-192*n^2+176*n-48)*x-(288*n^3-1008*n^2+1080*n-324)*x^2)*P(n-2,x) + ((64*n^3-192*n^2+176*n-48)*x^2+(128*n^3-448*n^2+480*n-144)*x^3)*d/dx P(n-2,x) + (8*n^2-12*n+2+(32*n^2-64*n+24)*x)*P(n-1,x) - (32*n^2-64*n+24)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (64*n^3 - 192*n^2 + 176*n - 48)*x - (288*n^3 - 1008*n^2 + 1080*n - 324)*x^2)*P[n - 2, x] + ((64*n^3 - 192*n^2 + 176*n - 48)*x^2 + (128*n^3 - 448*n^2 + 480*n - 144)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 12*n + 2 + (32*n^2 - 64*n + 24)*x)*P[n - 1, x] - (32*n^2 - 64*n + 24)*x^2*D[P[n - 1, x], x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (64*n^3 - 192*n^2 + 176*n - 48)*x - (288*n^3 - 1008*n^2 + 1080*n - 324)*x^2)*v[i - 2] + ((64*n^3 - 192*n^2 + 176*n - 48)*x^2 + (128*n^3 - 448*n^2 + 480*n - 144)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 12*n + 2 + (32*n^2 - 64*n + 24)*x)*v[i - 1] - (32*n^2 - 64*n + 24)*x^2*deriv(v[i - 1], x))/(2*n - 3)); 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-2b477f2aa77e3efc","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-17d7b4a637c00413","e-61db0af244b23600"],"text":"A109692 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A109692 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}