{"id":"n-a06d487b0a5cfe40","revision":0,"first_index":0,"fingerprint":"f6-8d44fb57f80b5d40","name":"theta_plus_one_squared of basis_map_A013611 of Triangle of coefficients in expansion of (1+4x)^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"],["5","64"],["25","640","2304"],["125","4800","34560","65536"],["625","32000","345600","1310720","1638400"],["3125","200000","2880000","16384000","40960000","37748736"],["15625","1200000","21600000","163840000","614400000","1132462080","822083584"],["78125","7000000","151200000","1433600000","7168000000","19818086400","28772925440","17179869184"],["390625","40000000","1008000000","11468800000","71680000000","264241152000","575458508800","687194767360","347892350976"],["1953125","225000000","6480000000","86016000000","645120000000","2972712960000","8631877632000","15461882265600","15655155793920","6871947673600"],["9765625","1250000000","40500000000","614400000000","5376000000000","29727129600000","107898470400000","257698037760000","391378894848000","343597383680000","133040906960896"],["48828125","6875000000","247500000000","4224000000000","42240000000000","272498688000000","1186883174400000","3543348019200000","7175279738880000","9448928051200000","7317249882849280","2533274790395904"],["244140625","37500000000","1485000000000","28160000000000","316800000000000","2335703040000000","11868831744000000","42520176230400000","107629196083200000","188978561024000000","219517496485478400","151996487423754240","47569271064100864"],["1220703125","203125000000","8775000000000","183040000000000","2288000000000000","18977587200000000","110210580480000000","460635242496000000","1399179549081600000","3070901616640000000","4756212423852032000","4939885841272012800","3092002619166556160","882705526964617216"],["6103515625","1093750000000","51187500000000","1164800000000000","16016000000000000","147603456000000000","964342579200000000","4606352424960000000","16323761405952000000","42992622632960000000","83233717417410560000","115264002963013632000","108220091670829465600","61789386887523205120","16212958658533785600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","69","2969","105021","3327345","98175861","2755601289","74530759149","1958983969761","50331419278245","1269372983014521","31524180768753309","772762205839074129","18733191006498521301","449772073183736163945"],"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"]],"recurrence":{"denominator":{"coeffs":[["25","-80","0"],["0","240","0"],["0","0","256"]]},"terms":[{"coeff":{"coeffs":[["125","1200","3840","4096"],["0","1200","6400","8192"],["0","0","1280","4096"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190991,"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-7a4c53b34be22b52"]},{"id":190992,"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-7a4c53b34be22b52"]},{"id":190993,"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-7a4c53b34be22b52"]},{"id":190994,"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-7a4c53b34be22b52"]}],"region_exact":{},"incoming_edges":[{"id":"e-7a4c53b34be22b52","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-a06d487b0a5cfe40","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":"be51b288c1c58c79ca2c2ce1c8bf03d6fe1a1726c6eee2dc2124f74ccb232dec","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-3deb3ea1ada5b4a1","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+4x)^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:       5,        64;\n  2:      25,       640,       2304;\n  3:     125,      4800,      34560,       65536;\n  4:     625,     32000,     345600,     1310720,      1638400;\n  5:    3125,    200000,    2880000,    16384000,     40960000,      37748736;\n  6:   15625,   1200000,   21600000,   163840000,    614400000,    1132462080,     822083584;\n  7:   78125,   7000000,  151200000,  1433600000,   7168000000,   19818086400,   28772925440,    17179869184;\n  8:  390625,  40000000, 1008000000, 11468800000,  71680000000,  264241152000,  575458508800,   687194767360,   347892350976;\n  9: 1953125, 225000000, 6480000000, 86016000000, 645120000000, 2972712960000, 8631877632000, 15461882265600, 15655155793920, 6871947673600;\n","recurrence_sentence":"The row polynomials satisfy (25+(240*n-80)*x+256*n^2*x^2)*P(n,x) = (125+(1200*n+1200)*x+(1280*n^2+6400*n+3840)*x^2+(4096*n^2+8192*n+4096)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((125 + (1200*n + 1200)*x + (1280*n^2 + 6400*n + 3840)*x^2 + (4096*n^2 + 8192*n + 4096)*x^3)*P[n - 1, x])/(25 + (240*n - 80)*x + 256*n^2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((125 + (1200*n + 1200)*x + (1280*n^2 + 6400*n + 3840)*x^2 + (4096*n^2 + 8192*n + 4096)*x^3)*v[i - 1])/(25 + (240*n - 80)*x + 256*n^2*x^2)); 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-2e0409c0014ec240","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-189a8da608edbb49","e-7a4c53b34be22b52"],"text":"A013611 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A013611 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}