{"id":"n-03fa9c0ed408d1dc","revision":0,"first_index":0,"fingerprint":"f6-3d72d99fe3345b32","name":"theta_plus_one_squared of Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","4"],["6","32","9"],["24","232","198","16"],["120","1776","2952","832","25"],["720","14832","39600","23232","2850","36"],["5040","135936","523260","513920","140250","8640","49"],["40320","1364544","7067736","10304320","4895000","718200","24206","64"],["362880","14935680","99236664","199041024","144137500","38250000","3295740","64256","81"],["3628800","177356160","1459682208","3822478464","3883934600","1611540000","260981840","14006272","164106","100"],["39916800","2273425920","22567330944","74274623232","100067377200","59341834944","15404116280","1615609856","56298078","407200","121"],["479001600","31310876160","367094716416","1475196643584","2529510825600","2017490356224","761899078416","131294897664","9304986456","217062600","988086","144"],["6227020800","461344343040","6281364931200","30129274589184","63746636835600","65454791122944","33698282272816","8596698894336","1026255239856","50725927600","810663216","2355264","169"],["87178291200","7243970227200","112958136161280","635033248892928","1618209378496800","2069797444725312","1386695593239488","488629257561088","87858797505696","7501605222800","264969406768","2953150272","5532722","196"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","6","47","470","5705","81270","1327095","24414390","499323825","11233772550","275640940575","7323799432950","209452414196025","6414282538413750"],"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"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-16","4"],["2","24","-28"],["-5","-24","53"],["4","16","-38"],["-1","-4","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-11","5"],["0","0","-2","-32"],["0","3","16","53"],["0","-2","-12","-30"],["0","0","4","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","2","-3","1"],["0","0","0","6","-6"],["0","0","-1","-8","9"],["0","0","0","4","-4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4"],["3","12"],["-5","-12"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","0"],["0","-2","-4"],["0","2","8"],["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181060,"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-3f542f4b4c5c48e3"]},{"id":181061,"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-3f542f4b4c5c48e3"]},{"id":181062,"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-3f542f4b4c5c48e3"]},{"id":181063,"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-3f542f4b4c5c48e3"]}],"region_exact":{},"incoming_edges":[{"id":"e-3f542f4b4c5c48e3","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-03fa9c0ed408d1dc","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":"efdeff9cca4fc5eee0a41da5ad8390317ca2ecfa6dee29b1f89b092170870f71","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-e14e37227fef094e","role":"sequence","name":"Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,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":[{"id":"l-9621f0ba1c7acdd8","kind":"fingerprint_related","node_a_id":"n-03fa9c0ed408d1dc","node_b_id":"n-6bbfb631a2c1156c","fingerprint":"f6-3d72d99fe3345b32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3d72d99fe3345b32"},"other_id":"n-6bbfb631a2c1156c","other":{"id":"n-6bbfb631a2c1156c","name":"theta_plus_one_squared of Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.","first_index":0,"fingerprint":"f6-3d72d99fe3345b32","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["6","32","9"]],"card_rows":[["1"],["2","4"],["6","32","9"],["24","232","198","16"]],"row_sums_preview":["1","6","47","470","5705","81270","1327095","24414390"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9b757b008ff4762d","steps":["(\u03b8+1)\u00b2"],"edges":["e-8733b22e73c455f4"],"text":"A112007 \u2192 (\u03b8+1)\u00b2"},"headline":"A112007 \u2192 (\u03b8+1)\u00b2","chain":null}}],"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;\n  2:      2,        4;\n  3:      6,       32,        9;\n  4:     24,      232,      198,        16;\n  5:    120,     1776,     2952,       832,        25;\n  6:    720,    14832,    39600,     23232,      2850,       36;\n  7:   5040,   135936,   523260,    513920,    140250,     8640,      49;\n  8:  40320,  1364544,  7067736,  10304320,   4895000,   718200,   24206,    64;\n  9: 362880, 14935680, 99236664, 199041024, 144137500, 38250000, 3295740, 64256, 81;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^4-4*n^3+5*n^2-2*n+(4*n^4-16*n^3+24*n^2-24*n+16)*x-(9*n^4-38*n^3+53*n^2-28*n+4)*x^2)*P(n-2,x) - ((2*n^3-3*n^2)*x-(4*n^4-12*n^3+16*n^2-2*n-11)*x^2-(4*n^4-30*n^3+53*n^2-32*n+5)*x^3)*d/dx P(n-2,x) - ((n^2-2)*x^2-(4*n^3-8*n^2+6*n-3)*x^3+(4*n^3-9*n^2+6*n-1)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3-5*n^2+3*n+(4*n^3-12*n^2+12*n-4)*x)*P(n-1,x) + ((2*n^2-2*n-1)*x-(4*n^3-8*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 4*n^3 + 5*n^2 - 2*n + (4*n^4 - 16*n^3 + 24*n^2 - 24*n + 16)*x - (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*P[n - 2, x] - ((2*n^3 - 3*n^2)*x - (4*n^4 - 12*n^3 + 16*n^2 - 2*n - 11)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*D[P[n - 2, x], x] - ((n^2 - 2)*x^2 - (4*n^3 - 8*n^2 + 6*n - 3)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 - 5*n^2 + 3*n + (4*n^3 - 12*n^2 + 12*n - 4)*x)*P[n - 1, x] + ((2*n^2 - 2*n - 1)*x - (4*n^3 - 8*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 - 4*n^3 + 5*n^2 - 2*n + (4*n^4 - 16*n^3 + 24*n^2 - 24*n + 16)*x - (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*v[i - 2] - ((2*n^3 - 3*n^2)*x - (4*n^4 - 12*n^3 + 16*n^2 - 2*n - 11)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*deriv(v[i - 2], x) - ((n^2 - 2)*x^2 - (4*n^3 - 8*n^2 + 6*n - 3)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 - 5*n^2 + 3*n + (4*n^3 - 12*n^2 + 12*n - 4)*x)*v[i - 1] + ((2*n^2 - 2*n - 1)*x - (4*n^3 - 8*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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-e14e37227fef094e","steps":["(\u03b8+1)\u00b2"],"edges":["e-3f542f4b4c5c48e3"],"text":"A163936 \u2192 (\u03b8+1)\u00b2"},"headline":"A163936 \u2192 (\u03b8+1)\u00b2","chain":null}
