{"id":"n-f6cc84113892da54","revision":0,"first_index":1,"fingerprint":"f6-f275e5e9925d5c1c","name":"theta_plus_one_squared of reversal of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","12"],["1","20","45"],["1","28","126","112"],["1","36","243","480","225"],["1","44","396","1232","1375","396"],["1","52","585","2496","4550","3276","637"],["1","60","810","4400","11250","13608","6860","960"],["1","68","1071","7072","23375","40392","34986","13056","1377"],["1","76","1368","10640","43225","97812","122892","80256","23085","1900"],["1","84","1701","15232","73500","206388","343343","329472","168399","38500","2541"],["1","92","2070","20976","117300","394128","820456","1052480","799227","328900","61226","3312"],["1","100","2475","28000","178125","697680","1749300","2828800","2895750","1787500","605605","93600","4225"],["1","108","2916","36432","259875","1163484","3418632","6697728","8699886","7293000","3737448","1061424","138411","5292"],["1","116","3393","46400","366850","1848924","6236769","14387712","22761810","24354200","17060758","7383168","1783964","198940","6525"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","66","267","985","3444","11597","37949","121398","381255","1179161","3600168","10871161","32514637","96439530"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-12","66","-12"],["32","-254","80"],["-28","356","-204"],["8","-216","248"],["0","48","-144"],["0","0","32"]]},"terms":[{"coeff":{"coeffs":[["-12","-78","-132","-48"],["32","130","-8","-40"],["-28","20","544","252"],["8","-120","-536","-88"],["0","48","96","-176"],["0","0","32","96"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-27","-50","-24"],["0","8","30","-16","-32"],["0","-4","20","172","112"],["0","0","-24","-96","0"],["0","0","0","-16","-64"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173449,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-d15ff524a14f8758"]},{"id":173450,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-d15ff524a14f8758"]},{"id":173451,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-d15ff524a14f8758"]}],"region_exact":{},"incoming_edges":[{"id":"e-d15ff524a14f8758","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-f6cc84113892da54","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":"9825594c45689d13afd240f603abb33c2aa89f64857a0e9b6855772bf93884b2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7895ffdfea7c95d5","role":"sequence","name":"reversal of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-038960540e5f9eaf","kind":"fingerprint_related","node_a_id":"n-bbda111424d81fcc","node_b_id":"n-f6cc84113892da54","fingerprint":"f6-f275e5e9925d5c1c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f275e5e9925d5c1c"},"other_id":"n-bbda111424d81fcc","other":{"id":"n-bbda111424d81fcc","name":"theta_plus_one_squared of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","first_index":0,"fingerprint":"f6-f275e5e9925d5c1c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","12"],["1","20","45"]],"card_rows":[["1"],["1","12"],["1","20","45"],["1","28","126","112"]],"row_sums_preview":["1","13","66","267","985","3444","11597","37949"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9496336e2b29e28e","steps":["(\u03b8+1)\u00b2"],"edges":["e-1c39fb0703d12b2d"],"text":"A082985 \u2192 (\u03b8+1)\u00b2"},"headline":"A082985 \u2192 (\u03b8+1)\u00b2","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1, 12;\n   3: 1, 20,   45;\n   4: 1, 28,  126,   112;\n   5: 1, 36,  243,   480,   225;\n   6: 1, 44,  396,  1232,  1375,   396;\n   7: 1, 52,  585,  2496,  4550,  3276,    637;\n   8: 1, 60,  810,  4400, 11250, 13608,   6860,   960;\n   9: 1, 68, 1071,  7072, 23375, 40392,  34986, 13056,  1377;\n  10: 1, 76, 1368, 10640, 43225, 97812, 122892, 80256, 23085, 1900;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-28*n^2+32*n-12+(48*n^4-216*n^3+356*n^2-254*n+66)*x+(32*n^5-144*n^4+248*n^3-204*n^2+80*n-12)*x^2)*P(n,x) = (8*n^3-28*n^2+32*n-12+(48*n^4-120*n^3+20*n^2+130*n-78)*x+(32*n^5+96*n^4-536*n^3+544*n^2-8*n-132)*x^2+(96*n^5-176*n^4-88*n^3+252*n^2-40*n-48)*x^3)*P(n-1,x) - ((4*n^2-8*n+4)*x+(24*n^3-20*n^2-30*n+27)*x^2+(16*n^4+96*n^3-172*n^2+16*n+50)*x^3+(64*n^4-112*n^2+32*n+24)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 120*n^3 + 20*n^2 + 130*n - 78)*x + (32*n^5 + 96*n^4 - 536*n^3 + 544*n^2 - 8*n - 132)*x^2 + (96*n^5 - 176*n^4 - 88*n^3 + 252*n^2 - 40*n - 48)*x^3)*P[n - 1, x] - ((4*n^2 - 8*n + 4)*x + (24*n^3 - 20*n^2 - 30*n + 27)*x^2 + (16*n^4 + 96*n^3 - 172*n^2 + 16*n + 50)*x^3 + (64*n^4 - 112*n^2 + 32*n + 24)*x^4)*D[P[n - 1, x], x])/(8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 216*n^3 + 356*n^2 - 254*n + 66)*x + (32*n^5 - 144*n^4 + 248*n^3 - 204*n^2 + 80*n - 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 120*n^3 + 20*n^2 + 130*n - 78)*x + (32*n^5 + 96*n^4 - 536*n^3 + 544*n^2 - 8*n - 132)*x^2 + (96*n^5 - 176*n^4 - 88*n^3 + 252*n^2 - 40*n - 48)*x^3)*v[i - 1] - ((4*n^2 - 8*n + 4)*x + (24*n^3 - 20*n^2 - 30*n + 27)*x^2 + (16*n^4 + 96*n^3 - 172*n^2 + 16*n + 50)*x^3 + (64*n^4 - 112*n^2 + 32*n + 24)*x^4)*deriv(v[i - 1], x))/(8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 216*n^3 + 356*n^2 - 254*n + 66)*x + (32*n^5 - 144*n^4 + 248*n^3 - 204*n^2 + 80*n - 12)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-2020a36645e8b930","e-d15ff524a14f8758"],"text":"A111125 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A111125 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}
