{"id":"n-d5996ebbf963d51d","revision":0,"first_index":0,"fingerprint":"f6-9b2962a65732d26e","name":"theta_plus_one_squared of basis_map_A013610 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of 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","12"],["8","72","81"],["48","528","972","432"],["384","4800","11340","8640","2025"],["3840","52608","145800","146880","60750","8748"],["46080","677376","2104704","2540160","1417500","367416","35721"],["645120","10036224","34038144","46787328","31752000","11267424","2000376","139968"],["10321920","168321024","612354816","930134016","727347600","317447424","78014664","10077696","531441"],["185794560","3153641472","12158553600","20008304640","17452260000","8856195264","2700507600","487088640","47829690","1968300"],["3715891200","65302364160","264458151936","465011712000","442834128000","252517668480","90172949328","20323353600","2806008480","216513000","7144929"],["81749606400","1481242705920","6258870300672","11640701140992","11922093216000","7468432139520","3014918698176","800444519424","138897419760","15155910000","943130628","25509168"],["1961990553600","36530820218880","160211275481088","312757468987392","340696023782400","230745814041600","102854146659264","31024227446784","6372725858928","878176728000","77651088372","3979430208","89813529"],["51011754393600","973345212334080","4412076198985728","8986154329571328","10324147254220800","7471197987821568","3616419887144064","1209650243383296","283485985919136","46435283294400","5206709820312","380698823232","16346062278","312487308"],["1428329123020800","27865806998077440","130108213754855424","275143394289254400","331198721538048000","253793859797237760","131767148623638528","48040668413337600","12530498373581760","2350136028240000","314347953327408","29248812028800","1798066850580","65622334680","1076168025"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","14","161","1980","27189","418626","7188957","136666584","2854550601","65052143766","1607365885113","42823474296660","1224114390090045","37379528204261130","1214572238470001205"],"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","12"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-4","24","81"],["8","-24","0"],["-4","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-24","36"],["0","0","24","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","12"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-12"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186803,"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-661b6e89217784d0"]},{"id":186804,"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-661b6e89217784d0"]},{"id":186805,"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-661b6e89217784d0"]},{"id":186806,"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-661b6e89217784d0"]}],"region_exact":{},"incoming_edges":[{"id":"e-661b6e89217784d0","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-d5996ebbf963d51d","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":"08403e39159d33001553b9654986192b6b1054058aba5ff05d78d1dc29222a31","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-dd4c9e25ea6d895a","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of 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,         12;\n  2:         8,         72,          81;\n  3:        48,        528,         972,         432;\n  4:       384,       4800,       11340,        8640,        2025;\n  5:      3840,      52608,      145800,      146880,       60750,       8748;\n  6:     46080,     677376,     2104704,     2540160,     1417500,     367416,      35721;\n  7:    645120,   10036224,    34038144,    46787328,    31752000,   11267424,    2000376,    139968;\n  8:  10321920,  168321024,   612354816,   930134016,   727347600,  317447424,   78014664,  10077696,   531441;\n  9: 185794560, 3153641472, 12158553600, 20008304640, 17452260000, 8856195264, 2700507600, 487088640, 47829690, 1968300;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-8*n+4+(24*n-24)*x-81*x^2)*P(n-2,x) + ((24*n-24)*x^2+36*x^3)*d/dx P(n-2,x) + (4*n-2+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 8*n + 4 + (24*n - 24)*x - 81*x^2)*P[n - 2, x] + ((24*n - 24)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (4*n - 2 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 8*n + 4 + (24*n - 24)*x - 81*x^2)*v[i - 2] + ((24*n - 24)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (4*n - 2 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], 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-4f6e747b794bcc2e","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-45a2b063eb896a2b","e-661b6e89217784d0"],"text":"A028338 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A028338 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}