{"id":"n-eed5782960f78ff2","revision":0,"first_index":0,"fingerprint":"f6-c5aad9b6ec0d044d","name":"theta_plus_one_squared of basis_map_A013610 of Triangle of coefficients in expansion of (1+6x)^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"],["7","72"],["49","1008","2916"],["343","10584","61236","93312"],["2401","98784","857304","2612736","2624400"],["16807","864360","10001880","45722880","91854000","68024448"],["117649","7260624","105019740","640120320","1928934000","2857026816","1666598976"],["823543","59295096","1029193452","7841473920","31505922000","69997156992","81663349824","39182082048"],["5764801","474360768","9605805552","87824507904","441082908000","1306613597184","2286573795072","2194196594688","892616806656"],["40353607","3735591048","86452249968","922157332992","5557644640800","20579164155648","48018049696512","69117192732672","56234858819328","19835929036800"],["282475249","29054597040","756457187220","9221573329920","64839187476000","288108298179072","840315869688960","1612734497095680","1968220058676480","1388515032576000","432026534421504"],["1977326743","223720397208","6471911490660","88757643300480","713231062236000","3697389826631424","12940864393209984","31045139069091840","50517648172696320","53457828754176000","33266043150455808","9254651051409408"],["13841287201","1708410305952","54364056521544","828404670804480","7488926153478000","44368677919577088","181172101504939776","521558336360742912","1060870611626622720","1496819205116928000","1397173812319143936","777390688318390272","195504503461023744"],["96889010407","12955444820136","449739013041864","7538482504320768","75721364440722000","504693711335189376","2355237319564217088","7910301434804600832","19307845131604533504","34052636916410112000","42380938973680699392","35371276318486757376","17790909814953160704","4081301113671548928"],["678223072849","97664122490256","3672868606508556","67161025947585024","742069371519075600","5495553745649839872","28851657164661659328","110744220087264411648","315361470482874047232","667431683561638195200","1038333004855177135104","1155461693070567407616","871754580932704874496","399967509139811794944","84333007705968230400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","79","3973","165475","6195625","216484375","7205078125","231279296875","7218994140625","220355224609375","6604766845703125","194988250732421875","5683231353759765625","163838863372802734375","4678547382354736328125"],"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":[["49","-126","0"],["0","378","0"],["0","0","324"]]},"terms":[{"coeff":{"coeffs":[["343","2646","6804","5832"],["0","2646","11340","11664"],["0","0","2268","5832"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191759,"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-e7d4aa4152ef7106"]},{"id":191760,"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-e7d4aa4152ef7106"]},{"id":191761,"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-e7d4aa4152ef7106"]},{"id":191762,"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-e7d4aa4152ef7106"]}],"region_exact":{},"incoming_edges":[{"id":"e-e7d4aa4152ef7106","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-eed5782960f78ff2","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":"2d0716f2fa8f211a25a40cafda6b84a4d3bd543b4db199a24af0440e86ba82d5","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-b4e192fec3e320c1","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+6x)^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:        7,         72;\n  2:       49,       1008,        2916;\n  3:      343,      10584,       61236,        93312;\n  4:     2401,      98784,      857304,      2612736,       2624400;\n  5:    16807,     864360,    10001880,     45722880,      91854000,       68024448;\n  6:   117649,    7260624,   105019740,    640120320,    1928934000,     2857026816,     1666598976;\n  7:   823543,   59295096,  1029193452,   7841473920,   31505922000,    69997156992,    81663349824,    39182082048;\n  8:  5764801,  474360768,  9605805552,  87824507904,  441082908000,  1306613597184,  2286573795072,  2194196594688,   892616806656;\n  9: 40353607, 3735591048, 86452249968, 922157332992, 5557644640800, 20579164155648, 48018049696512, 69117192732672, 56234858819328, 19835929036800;\n","recurrence_sentence":"The row polynomials satisfy (49+(378*n-126)*x+324*n^2*x^2)*P(n,x) = (343+(2646*n+2646)*x+(2268*n^2+11340*n+6804)*x^2+(5832*n^2+11664*n+5832)*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[((343 + (2646*n + 2646)*x + (2268*n^2 + 11340*n + 6804)*x^2 + (5832*n^2 + 11664*n + 5832)*x^3)*P[n - 1, x])/(49 + (378*n - 126)*x + 324*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] = ((343 + (2646*n + 2646)*x + (2268*n^2 + 11340*n + 6804)*x^2 + (5832*n^2 + 11664*n + 5832)*x^3)*v[i - 1])/(49 + (378*n - 126)*x + 324*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-b7cf68162d40a956","steps":["x \u21a6 x/(x+1)","(\u03b8+1)\u00b2"],"edges":["e-149dcf3d57e14e87","e-e7d4aa4152ef7106"],"text":"A038277 \u2192 x \u21a6 x/(x+1) \u2192 (\u03b8+1)\u00b2"},"headline":"A038277 \u2192 x \u21a6 x/(x+1) \u2192 (\u03b8+1)\u00b2","chain":null}