{"id":"n-72c8c1d362d8a9d8","revision":0,"first_index":1,"fingerprint":"f6-67545869359869df","name":"theta_plus_one_squared of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","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"],["3","8"],["13","64","36"],["63","456","540","128"],["321","3072","5616","3072","400"],["1683","20040","49680","43520","14000","1152"],["8989","128064","401220","486400","258000","55296","3136"],["48639","806792","3056508","4744320","3542000","1282176","197568","8192"],["265729","5029888","22357440","42348544","40936000","21159936","5619712","655360","20736"],["1462563","31109256","158718528","355117056","422251200","285368832","109584384","22413312","2052864","51200"],["8097453","191200320","1101206340","2842767360","4016712000","3362881536","1703036160","509214720","83047680","6144000","123904"],["45046719","1169171784","7502808060","21954768000","35959704000","35968762368","22708992768","9023815680","2172614400","290048000","17718272","294912"],["251595969","7119424512","50371195248","164781788160","307083546000","357574238208","271035899904","135042564096","43510452480","8650752000","964964352","49545216","692224"],["1409933619","43199769288","334077728400","1208299682304","2525103438000","3356908334208","2973619224576","1786704691200","725599150848","194278656000","32530747392","3082420224","134983680","1605632"],["7923848253","261345892416","2193108774948","8690299677696","20130515605200","30095662012416","30533299148352","21516760252416","10580891473152","3587994009600","813747078144","116579893248","9511849984","359661568","3686400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","113","1187","12481","130075","1341105","13686195","138393345","1388129195","13824431473","136813744963","1346436658369","13184950365371","128538002863793"],"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":[["-4","21","-45","24","-4"],["4","-41","125","-100","20"],["0","20","-116","152","-40"],["0","0","36","-100","40"],["0","0","0","24","-20"],["0","0","0","0","4"]]},"terms":[{"coeff":{"coeffs":[["-44","183","105","-112","4","0"],["28","-383","323","556","-180","-32"],["0","168","-728","-276","596","40"],["0","0","300","-320","-516","40"],["0","0","0","184","84","-72"],["0","0","0","0","28","24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","18","48","-12","-40","0","0"],["0","-24","14","182","80","-80","-16"],["0","0","-60","-100","128","184","16"],["0","0","0","-48","-136","-72","16"],["0","0","0","0","-8","-24","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183249,"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-2c8d9c4af7933980"]},{"id":183250,"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-2c8d9c4af7933980"]},{"id":183251,"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-2c8d9c4af7933980"]}],"region_exact":{},"incoming_edges":[{"id":"e-2c8d9c4af7933980","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-72c8c1d362d8a9d8","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":"a72abb8523a632584fe8887f643228b5ccd990d787766f8c367deeb8b955ba5c","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-60c7f085d1907563","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","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":[],"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:       3,        8;\n   3:      13,       64,        36;\n   4:      63,      456,       540,       128;\n   5:     321,     3072,      5616,      3072,       400;\n   6:    1683,    20040,     49680,     43520,     14000,      1152;\n   7:    8989,   128064,    401220,    486400,    258000,     55296,      3136;\n   8:   48639,   806792,   3056508,   4744320,   3542000,   1282176,    197568,     8192;\n   9:  265729,  5029888,  22357440,  42348544,  40936000,  21159936,   5619712,   655360,   20736;\n  10: 1462563, 31109256, 158718528, 355117056, 422251200, 285368832, 109584384, 22413312, 2052864, 51200;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4+(20*n^2-41*n+21)*x+(36*n^3-116*n^2+125*n-45)*x^2+(24*n^4-100*n^3+152*n^2-100*n+24)*x^3+(4*n^5-20*n^4+40*n^3-40*n^2+20*n-4)*x^4)*P(n,x) = (28*n-44+(168*n^2-383*n+183)*x+(300*n^3-728*n^2+323*n+105)*x^2+(184*n^4-320*n^3-276*n^2+556*n-112)*x^3+(28*n^5+84*n^4-516*n^3+596*n^2-180*n+4)*x^4+(24*n^5-72*n^4+40*n^3+40*n^2-32*n)*x^5)*P(n-1,x) - (2+(24*n-18)*x+(60*n^2-14*n-48)*x^2+(48*n^3+100*n^2-182*n+12)*x^3+(8*n^4+136*n^3-128*n^2-80*n+40)*x^4+(24*n^4+72*n^3-184*n^2+80*n)*x^5+(16*n^4-16*n^3-16*n^2+16*n)*x^6)*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[((28*n - 44 + (168*n^2 - 383*n + 183)*x + (300*n^3 - 728*n^2 + 323*n + 105)*x^2 + (184*n^4 - 320*n^3 - 276*n^2 + 556*n - 112)*x^3 + (28*n^5 + 84*n^4 - 516*n^3 + 596*n^2 - 180*n + 4)*x^4 + (24*n^5 - 72*n^4 + 40*n^3 + 40*n^2 - 32*n)*x^5)*P[n - 1, x] - (2 + (24*n - 18)*x + (60*n^2 - 14*n - 48)*x^2 + (48*n^3 + 100*n^2 - 182*n + 12)*x^3 + (8*n^4 + 136*n^3 - 128*n^2 - 80*n + 40)*x^4 + (24*n^4 + 72*n^3 - 184*n^2 + 80*n)*x^5 + (16*n^4 - 16*n^3 - 16*n^2 + 16*n)*x^6)*D[P[n - 1, x], x])/(4*n - 4 + (20*n^2 - 41*n + 21)*x + (36*n^3 - 116*n^2 + 125*n - 45)*x^2 + (24*n^4 - 100*n^3 + 152*n^2 - 100*n + 24)*x^3 + (4*n^5 - 20*n^4 + 40*n^3 - 40*n^2 + 20*n - 4)*x^4)]];\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] = ((28*n - 44 + (168*n^2 - 383*n + 183)*x + (300*n^3 - 728*n^2 + 323*n + 105)*x^2 + (184*n^4 - 320*n^3 - 276*n^2 + 556*n - 112)*x^3 + (28*n^5 + 84*n^4 - 516*n^3 + 596*n^2 - 180*n + 4)*x^4 + (24*n^5 - 72*n^4 + 40*n^3 + 40*n^2 - 32*n)*x^5)*v[i - 1] - (2 + (24*n - 18)*x + (60*n^2 - 14*n - 48)*x^2 + (48*n^3 + 100*n^2 - 182*n + 12)*x^3 + (8*n^4 + 136*n^3 - 128*n^2 - 80*n + 40)*x^4 + (24*n^4 + 72*n^3 - 184*n^2 + 80*n)*x^5 + (16*n^4 - 16*n^3 - 16*n^2 + 16*n)*x^6)*deriv(v[i - 1], x))/(4*n - 4 + (20*n^2 - 41*n + 21)*x + (36*n^3 - 116*n^2 + 125*n - 45)*x^2 + (24*n^4 - 100*n^3 + 152*n^2 - 100*n + 24)*x^3 + (4*n^5 - 20*n^4 + 40*n^3 - 40*n^2 + 20*n - 4)*x^4)); 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-78ba05b4a3dd925d","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-2f2f885dba2c5644","e-2c8d9c4af7933980"],"text":"A104684 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A104684 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}