{"id":"n-3b76d56c9dfaad44","revision":0,"first_index":1,"fingerprint":"f6-980e267510c7a2f6","name":"theta_plus_one of derivative 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":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["6","4"],["30","48","9"],["140","360","180","16"],["630","2240","1890","480","25"],["2772","12600","15120","6720","1050","36"],["12012","66528","103950","67200","18900","2016","49"],["51480","336336","648648","554400","231000","45360","3528","64"],["218790","1647360","3783780","4036032","2252250","665280","97020","5760","81"],["923780","7876440","21003840","26906880","18918900","7567560","1681680","190080","8910","100"],["3879876","36951200","112239270","168030720","142942800","72648576","22072050","3843840","347490","13200","121"],["16224936","170714544","581981400","997682400","997682400","617512896","240143904","57657600","8108100","600600","18876","144"],["67603900","778796928","2944825884","5690484800","6547290750","4788875520","2281367088","705729024","137837700","16016000","990990","26208","169"],["280816200","3515402800","14602442400","31411476096","40900359500","34569695160","19554575040","7449361920","1885619736","306306000","29949920","1572480","35490","196"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","10","87","696","5265","38298","270655","1870816","12706353","85078170","562969143","3688327800","23959844961","154507612938"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-12","16"],["20","-32"],["-11","24"],["2","-8"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["-36","-18","16"],["66","12","-46"],["-40","18","47"],["8","-18","-20"],["0","4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-9","-5","4"],["0","12","2","-10"],["0","-4","4","8"],["0","0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178472,"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-ab8c7d1711217003"]},{"id":178473,"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-ab8c7d1711217003"]},{"id":178474,"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-ab8c7d1711217003"]}],"region_exact":{},"incoming_edges":[{"id":"e-ab8c7d1711217003","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-3b76d56c9dfaad44","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":"df0553964dee8c5bf358004a2bdeaf20fccbb7ca2640aff92927c0b3316228b1","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-1ff7b1d8c09df87e","role":"sequence","name":"derivative 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:      0;\n   2:      1;\n   3:      6,       4;\n   4:     30,      48,       9;\n   5:    140,     360,     180,      16;\n   6:    630,    2240,    1890,     480,      25;\n   7:   2772,   12600,   15120,    6720,    1050,     36;\n   8:  12012,   66528,  103950,   67200,   18900,   2016,    49;\n   9:  51480,  336336,  648648,  554400,  231000,  45360,  3528,   64;\n  10: 218790, 1647360, 3783780, 4036032, 2252250, 665280, 97020, 5760, 81;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-11*n^2+20*n-12+(n^4-8*n^3+24*n^2-32*n+16)*x)*P(n,x) = (8*n^3-40*n^2+66*n-36+(4*n^4-18*n^3+18*n^2+12*n-18)*x+(3*n^4-20*n^3+47*n^2-46*n+16)*x^2)*P(n-1,x) - ((4*n^2-12*n+9)*x+(2*n^3-4*n^2-2*n+5)*x^2+(2*n^3-8*n^2+10*n-4)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 40*n^2 + 66*n - 36 + (4*n^4 - 18*n^3 + 18*n^2 + 12*n - 18)*x + (3*n^4 - 20*n^3 + 47*n^2 - 46*n + 16)*x^2)*P[n - 1, x] - ((4*n^2 - 12*n + 9)*x + (2*n^3 - 4*n^2 - 2*n + 5)*x^2 + (2*n^3 - 8*n^2 + 10*n - 4)*x^3)*D[P[n - 1, x], x])/(2*n^3 - 11*n^2 + 20*n - 12 + (n^4 - 8*n^3 + 24*n^2 - 32*n + 16)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^3 - 40*n^2 + 66*n - 36 + (4*n^4 - 18*n^3 + 18*n^2 + 12*n - 18)*x + (3*n^4 - 20*n^3 + 47*n^2 - 46*n + 16)*x^2)*v[i - 1] - ((4*n^2 - 12*n + 9)*x + (2*n^3 - 4*n^2 - 2*n + 5)*x^2 + (2*n^3 - 8*n^2 + 10*n - 4)*x^3)*deriv(v[i - 1], x))/(2*n^3 - 11*n^2 + 20*n - 12 + (n^4 - 8*n^3 + 24*n^2 - 32*n + 16)*x)); 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":["D","\u03b8+1"],"edges":["e-4c8816dc3f762029","e-ab8c7d1711217003"],"text":"A104684 \u2192 D \u2192 \u03b8+1"},"headline":"A104684 \u2192 D \u2192 \u03b8+1","chain":null}