{"id":"n-359a01e1c9d099c3","revision":0,"first_index":0,"fingerprint":"f6-f8028a91b50975db","name":"theta_plus_one_squared of Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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"],["0","4"],["0","4","9"],["0","4","27","16"],["0","4","45","96","25"],["0","4","63","240","250","36"],["0","4","81","448","875","540","49"],["0","4","99","720","2100","2520","1029","64"],["0","4","117","1056","4125","7560","6174","1792","81"],["0","4","135","1456","7150","17820","22638","13440","2916","100"],["0","4","153","1920","11375","36036","63063","59136","26730","4500","121"],["0","4","171","2448","17000","65520","147147","192192","138996","49500","6655","144"],["0","4","189","3040","24225","110160","303212","512512","521235","300300","86515","9504","169"],["0","4","207","3696","33250","174420","569772","1188096","1575288","1287000","605605","144144","13182","196"],["0","4","225","4416","44275","263340","997101","2480640","4081428","4375800","2941510","1153152","230685","17836","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","13","47","170","593","1997","6536","20909","65659","203038","619777","1871065","5594860","16590637"],"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":[["12","-12","0"],["-32","74","0"],["16","-104","12"],["0","40","-32"],["0","0","16"]]},"terms":[{"coeff":{"coeffs":[["8","58","128","36"],["-24","-138","-132","-48"],["16","24","-164","-100"],["0","40","144","48"],["0","0","16","48"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","29","58","24"],["0","-8","-40","-36","-16"],["0","0","-20","-100","-80"],["0","0","0","-8","-32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172135,"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-68a2eb61a78b6c20"]},{"id":172136,"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-68a2eb61a78b6c20"]},{"id":172137,"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-68a2eb61a78b6c20"]},{"id":172138,"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-68a2eb61a78b6c20"]}],"region_exact":{},"incoming_edges":[{"id":"e-68a2eb61a78b6c20","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-359a01e1c9d099c3","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":"161e7fba5a06f49afa71addae4e046471464a24a89c6a0e24e62a320caca9afd","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-3af57aaf5c129e4b","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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: 0, 4;\n  2: 0, 4,   9;\n  3: 0, 4,  27,   16;\n  4: 0, 4,  45,   96,   25;\n  5: 0, 4,  63,  240,  250,    36;\n  6: 0, 4,  81,  448,  875,   540,    49;\n  7: 0, 4,  99,  720, 2100,  2520,  1029,    64;\n  8: 0, 4, 117, 1056, 4125,  7560,  6174,  1792,   81;\n  9: 0, 4, 135, 1456, 7150, 17820, 22638, 13440, 2916, 100;\n","recurrence_sentence":"The row polynomials satisfy (16*n^2-32*n+12+(40*n^3-104*n^2+74*n-12)*x+(16*n^4-32*n^3+12*n^2)*x^2)*P(n,x) = (16*n^2-24*n+8+(40*n^3+24*n^2-138*n+58)*x+(16*n^4+144*n^3-164*n^2-132*n+128)*x^2+(48*n^4+48*n^3-100*n^2-48*n+36)*x^3)*P(n-1,x) - ((8*n-4)*x+(20*n^2+40*n-29)*x^2+(8*n^3+100*n^2+36*n-58)*x^3+(32*n^3+80*n^2+16*n-24)*x^4)*d/dx 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[((16*n^2 - 24*n + 8 + (40*n^3 + 24*n^2 - 138*n + 58)*x + (16*n^4 + 144*n^3 - 164*n^2 - 132*n + 128)*x^2 + (48*n^4 + 48*n^3 - 100*n^2 - 48*n + 36)*x^3)*P[n - 1, x] - ((8*n - 4)*x + (20*n^2 + 40*n - 29)*x^2 + (8*n^3 + 100*n^2 + 36*n - 58)*x^3 + (32*n^3 + 80*n^2 + 16*n - 24)*x^4)*D[P[n - 1, x], x])/(16*n^2 - 32*n + 12 + (40*n^3 - 104*n^2 + 74*n - 12)*x + (16*n^4 - 32*n^3 + 12*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] = ((16*n^2 - 24*n + 8 + (40*n^3 + 24*n^2 - 138*n + 58)*x + (16*n^4 + 144*n^3 - 164*n^2 - 132*n + 128)*x^2 + (48*n^4 + 48*n^3 - 100*n^2 - 48*n + 36)*x^3)*v[i - 1] - ((8*n - 4)*x + (20*n^2 + 40*n - 29)*x^2 + (8*n^3 + 100*n^2 + 36*n - 58)*x^3 + (32*n^3 + 80*n^2 + 16*n - 24)*x^4)*deriv(v[i - 1], x))/(16*n^2 - 32*n + 12 + (40*n^3 - 104*n^2 + 74*n - 12)*x + (16*n^4 - 32*n^3 + 12*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-3af57aaf5c129e4b","steps":["(\u03b8+1)\u00b2"],"edges":["e-68a2eb61a78b6c20"],"text":"A121314 \u2192 (\u03b8+1)\u00b2"},"headline":"A121314 \u2192 (\u03b8+1)\u00b2","chain":null}